Navier–Stokes Equation Ontology

flow "NavierStokesEquationOntology" v1.0

# NAVIER-STOKES EQUATION ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/mathematical-physics-analysis/navier-stokes-equation-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Mathematical Physics Analysis"
ENTITY := "Navier-Stokes Equation"

NAVIER_STOKES_EQUATION := {
 type: NonlinearFluidDynamicsPDE,
 domain: MathematicalPhysicsAnalysis,
 purpose: "Represent conservation of momentum in viscous fluids together with mass conservation, pressure, body forces, constitutive laws and boundary conditions."
}

CONCEPTS := {
 NAVIER_STOKES_EQUATION,
 VELOCITY_FIELD,
 PRESSURE_FIELD,
 DENSITY,
 VISCOSITY,
 MOMENTUM_CONSERVATION,
 MASS_CONSERVATION,
 CONTINUITY_EQUATION,
 INCOMPRESSIBILITY,
 CONVECTIVE_TERM,
 VISCOUS_TERM,
 PRESSURE_GRADIENT,
 BODY_FORCE,
 REYNOLDS_NUMBER,
 VORTICITY,
 TURBULENCE
}

# ---------- INCOMPRESSIBLE FORM ----------
rho
*
(
 partial_u/partial_t
 +
(u dot grad)u
)
 =
-grad(p)
+
mu*LAPLACIAN(u)
+
rho*f

with

div(u)
 =
0

u
 -> VELOCITY_FIELD

p
 -> PRESSURE_FIELD

rho
 -> FLUID_DENSITY

mu
 -> DYNAMIC_VISCOSITY

f
 -> BODY_FORCE_PER_UNIT_MASS

# ---------- KINEMATIC VISCOSITY ----------
nu
 :=
mu/rho

for constant density:

partial_u/partial_t
+
(u dot grad)u
 =
-(1/rho)*grad(p)
+
nu*LAPLACIAN(u)
+
f

nu
 -> KINEMATIC_VISCOSITY

# ---------- CONTINUITY EQUATION ----------
general mass conservation:

partial_rho/partial_t
+
div(
 rho*u
)
 =
0

CONTINUITY_EQUATION
 -> CONSERVATION_OF_MASS

# ---------- INCOMPRESSIBLE CONDITION ----------
for constant-density incompressible flow:

div(u)
 =
0

INCOMPRESSIBILITY
 -> divergence_free_velocity

INCOMPRESSIBLE_NAVIER_STOKES
 -> momentum_equation_plus(DIVERGENCE_CONSTRAINT)

# ---------- MATERIAL DERIVATIVE ----------
D/Dt
 :=
partial/partial_t
+
u dot grad

therefore

Du/Dt
 =
partial_u/partial_t
+
(u dot grad)u

MATERIAL_DERIVATIVE
 -> rate_seen_by(MOVING_FLUID_PARTICLE)

# ---------- CONVECTIVE TERM ----------
(u dot grad)u

-> NONLINEAR_ADVECTION_OF_MOMENTUM

CONVECTIVE_TERM
 -> principal_non_linearity_of(NAVIER_STOKES)

# ---------- PRESSURE TERM ----------
-grad(p)

-> PRESSURE_FORCE_PER_UNIT_VOLUME

pressure
 -> acts_as(LAGRANGE_MULTIPLIER_FOR_INCOMPRESSIBILITY)
    in common incompressible formulations

PRESSURE_FIELD
 -> enforces(DIVERGENCE_FREE_CONSTRAINT)

# ---------- VISCOUS TERM ----------
for Newtonian incompressible constant-viscosity fluid:

mu*LAPLACIAN(u)

-> VISCOUS_MOMENTUM_DIFFUSION

VISCOSITY
 -> smooths(VELOCITY_GRADIENTS)

# ---------- CAUCHY MOMENTUM EQUATION ----------
rho*Du/Dt
 =
div(sigma)
+
rho*f

sigma
 -> CAUCHY_STRESS_TENSOR

for Newtonian fluid:

sigma
 =
-p*I
+
tau

tau
 -> VISCOUS_STRESS

NAVIER_STOKES
 -> constitutive_specialisation_of(CAUCHY_MOMENTUM_BALANCE)

# ---------- NEWTONIAN STRESS ----------
for compressible Newtonian fluid:

tau
 =
mu
*
(
 grad(u)
 +
grad(u)^T
)
+
lambda
*
div(u)
*
I

CONSTITUTIVE_RELATION
 -> links(STRESS_TO_RATE_OF_STRAIN)

# ---------- BOUNDARY CONDITIONS ----------
NO_SLIP:

u
 =
u_wall

on solid boundary

NO_PENETRATION:

u dot n
 =
u_wall dot n

TRACTION:

sigma*n
 =
t_boundary

BOUNDARY_CONDITIONS
 -> essential_to(FLOW_SOLUTION)

# ---------- INITIAL CONDITION ----------
u(x,0)
 =
u_0(x)

for incompressible flow:

div(u_0)
 =
0

with compatibility at boundaries

INITIAL_VELOCITY
 -> starts(FLOW_EVOLUTION)

# ---------- PRESSURE POISSON EQUATION ----------
take divergence of incompressible momentum equation

with
div(u)=0

-> pressure satisfies Poisson-type equation

schematically:

LAPLACIAN(p)
 =
-rho
*
div(
 (u dot grad)u
)
+
source_terms

PRESSURE_POISSON
 -> enforces(INCOMPRESSIBILITY)

# ---------- REYNOLDS NUMBER ----------
Re
 :=
rho*U*L/mu

=
U*L/nu

REYNOLDS_NUMBER
 -> ratio_of(INERTIAL_TO_VISCOUS_EFFECTS)

low Re
 -> viscous_dominated

high Re
 -> inertia_dominated_and_possible_transition_to_turbulence

# ---------- NONDIMENSIONAL FORM ----------
scale:

x
 =
L*x_star

u
 =
U*u_star

t
 =
(L/U)*t_star

p
 =
rho*U^2*p_star

then

partial_u_star/partial_t_star
+
(u_star dot grad_star)u_star
 =
-grad_star(p_star)
+
(1/Re)
*
LAPLACIAN_star(u_star)
+
f_star

NONDIMENSIONALISATION
 -> exposes(REYNOLDS_NUMBER)

# ---------- STOKES FLOW ----------
if

Re << 1

inertia negligible

then

-grad(p)
+
mu*LAPLACIAN(u)
+
rho*f
 =
0

with

div(u)=0

STOKES_EQUATIONS
 -> linear_viscous_flow_limit

# ---------- EULER EQUATIONS ----------
if

mu
 =
0

then

rho
*
(
 partial_u/partial_t
 +
(u dot grad)u
)
 =
-grad(p)
+
rho*f

EULER_EQUATIONS
 -> inviscid_fluid_limit

# ---------- VORTICITY ----------
omega
 :=
curl(u)

VORTICITY
 -> local_rotation_measure

take curl of momentum equation

-> VORTICITY_EQUATION

# ---------- INCOMPRESSIBLE VORTICITY EQUATION ----------
for constant-density incompressible flow:

partial_omega/partial_t
+
(u dot grad)omega
 =
(omega dot grad)u
+
nu*LAPLACIAN(omega)
+
curl(f)

VORTEX_STRETCHING
 :=
(omega dot grad)u

VORTEX_STRETCHING
 -> absent_in_strictly_2D_incompressible_flow

# ---------- TWO DIMENSIONS ----------
in 2D incompressible flow:

vorticity
 -> scalar_out_of_plane

equation:

omega_t
+
u dot grad(omega)
 =
nu*LAPLACIAN(omega)

without suitable forcing

2D_FLOW
 -> lacks(3D_VORTEX_STRETCHING_TERM)

# ---------- ENERGY ----------
kinetic energy:

E(t)
 :=
1/2
*
rho
*
integral_Omega(
 |u|^2
 dx
)

for suitable incompressible no-forcing/no-energy-input boundaries:

dE/dt
 =
-mu
*
integral_Omega(
 |grad(u)|^2
 dx
)

or equivalent strain-rate dissipation form

VISCOSITY
 -> DISSIPATES(KINETIC_ENERGY)

# ---------- ENERGY BALANCE ----------
with forcing and boundaries:

rate_of_kinetic_energy
 =
POWER_INPUT
-
VISCOUS_DISSIPATION
+
BOUNDARY_WORK_TERMS

ENERGY_METHOD
 -> supports(STABILITY_AND_EXISTENCE_ESTIMATES)

# ---------- LAMINAR FLOW ----------
LAMINAR_FLOW
 -> ordered_smooth_flow_regime

low_or_moderate_Re
 -> often_supports(LAMINAR_BEHAVIOUR)

but

FLOW_REGIME
 -> depends_on(GEOMETRY_DISTURBANCES_AND_BOUNDARIES)

# ---------- TURBULENCE ----------
TURBULENT_FLOW
 -> multiscale_irregular_dynamics

high Reynolds number
 -> may_support(TURBULENCE)

TURBULENCE
 -> characterised_by(
 STRONG_NONLINEARITY,
 VORTICAL_STRUCTURES,
 ENERGY_TRANSFER_ACROSS_SCALES,
 SENSITIVE_DYNAMICS
)

# ---------- ENERGY CASCADE ----------
in 3D turbulence:

energy
 -> injected_at(LARGE_SCALES)

-> transferred_to(SMALLER_SCALES)

-> dissipated_by(VISCOSITY)

KOLMOGOROV_THEORY
 -> statistical_description_of(INERTIAL_RANGE)
    under idealised assumptions

# ---------- BOUNDARY LAYER ----------
high Re near solid boundary:

thin region
 -> strong_velocity_gradient

BOUNDARY_LAYER
 -> viscosity_remains_important_near(WALL)

PRANDTL_BOUNDARY_LAYER
 -> asymptotic_reduction_for(HIGH_RE_FLOW)

# ---------- POISEUILLE FLOW ----------
steady pressure-driven flow
between plates or in pipe

-> viscous_balance_with(PRESSURE_GRADIENT)

velocity profile
 -> parabolic
    in standard laminar Newtonian geometries

POISEUILLE_FLOW
 -> exact_Navier_Stokes_solution

# ---------- COUETTE FLOW ----------
flow between moving parallel plates

steady incompressible Newtonian case

-> linear_velocity_profile
   under simple zero-pressure-gradient assumptions

COUETTE_FLOW
 -> shear_driven_exact_solution

# ---------- COMPRESSIBLE NAVIER-STOKES ----------
unknowns may include:

rho
u
p
T
energy

system requires:

MASS_CONSERVATION
MOMENTUM_CONSERVATION
ENERGY_CONSERVATION
EQUATION_OF_STATE
CONSTITUTIVE_LAWS

COMPRESSIBLE_FLOW
 -> density_not_constant

# ---------- EXISTENCE AND REGULARITY ----------
in 2D incompressible Navier-Stokes:
 -> strong_global_regularities_results_available
    under standard settings

in 3D:
 global_smooth_solution_for_arbitrary_smooth_data
 -> major_open_problem

NAVIER_STOKES_EXISTENCE_AND_SMOOTHNESS
 -> Millennium_Prize_problem

MATHEMATICAL_ANALYSIS
 -> distinguishes(WEAK_SOLUTIONS_AND_SMOOTH_SOLUTIONS)

# ---------- WEAK SOLUTION ----------
multiply momentum equation by
 divergence_free_test_function

integrate over domain

-> weak_form

LERAY_HOPF_WEAK_SOLUTION
 -> finite_energy_weak_solution
    in standard incompressible setting

WEAK_FORMULATION
 -> reduces(DERIVATIVE_REQUIREMENTS)

# ---------- NUMERICAL METHODS ----------
FINITE_DIFFERENCE
 -> grid_based_discretisation

FINITE_VOLUME
 -> conservation_based_discretisation

FINITE_ELEMENT
 -> variational_discretisation

SPECTRAL_METHOD
 -> high_order_basis_discretisation

CFD
 -> COMPUTATIONAL_FLUID_DYNAMICS

# ---------- PRESSURE-VELOCITY COUPLING ----------
in incompressible numerics:

velocity
and
pressure

-> coupled_by(DIVERGENCE_CONSTRAINT)

methods:
 PROJECTION_METHOD
 SIMPLE_FAMILY
 PRESSURE_CORRECTION
 MIXED_FINITE_ELEMENT

INF_SUP_CONDITION
 -> important_for(MIXED_VELOCITY_PRESSURE_FINITE_ELEMENTS)

# ---------- PROJECTION METHOD ----------
predict intermediate velocity

-> solve(PRESSURE_POISSON_EQUATION)

-> project_velocity_onto(DIVERGENCE_FREE_SPACE)

PROJECTION
 -> enforces(INCOMPRESSIBILITY)

# ---------- CFL AND DIFFUSIVE LIMITS ----------
explicit time integration:

Delta_t
 -> constrained_by(ADVECTION_CFL)

and often

Delta_t
 -> constrained_by(VISCOUS_DIFFUSION)

NUMERICAL_STABILITY
 -> depends_on(GRID_SPEED_VISCOSITY_AND_SCHEME)

# ---------- APPLICATIONS ----------
NAVIER_STOKES_EQUATION
 -> applied_to(AERODYNAMICS)
 -> applied_to(HYDRODYNAMICS)
 -> applied_to(WEATHER_AND_CLIMATE)
 -> applied_to(OCEANOGRAPHY)
 -> applied_to(BLOOD_FLOW)
 -> applied_to(PIPE_FLOW)
 -> applied_to(TURBOMACHINERY)
 -> applied_to(COMPUTATIONAL_FLUID_DYNAMICS)

# ---------- RELATION GRAPH ----------
MASS_CONSERVATION
 -> CONTINUITY_EQUATION

MOMENTUM_CONSERVATION
 + NEWTONIAN_STRESS
 -> NAVIER_STOKES_EQUATION

VELOCITY_FIELD
 -> generates(CONVECTIVE_MOMENTUM_TRANSPORT)

PRESSURE
 -> enforces(INCOMPRESSIBILITY)

VISCOSITY
 -> produces(MOMENTUM_DIFFUSION_AND_DISSIPATION)

REYNOLDS_NUMBER
 -> compares(INERTIA_AND_VISCOSITY)

VORTICITY
 -> describes(ROTATIONAL_FLOW_STRUCTURE)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "Navier-Stokes equation" ~= "momentum conservation equation for a viscous Newtonian fluid",
 "incompressible Navier-Stokes" ~= "Navier-Stokes momentum equation coupled to divergence-free velocity",
 "convective term" ~= "nonlinear transport of momentum by the fluid velocity",
 "viscous term" ~= "momentum diffusion generated by fluid viscosity",
 "Reynolds number" ~= "dimensionless ratio comparing inertial and viscous effects",
 "vorticity" ~= "curl of the velocity field measuring local rotational motion"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 incompressible_and_compressible_forms_must_not_be_conflated: true,
 constitutive_assumptions_are_required_to_obtain_Navier_Stokes_from_general_momentum_balance: true,
 pressure_boundary_initial_and_velocity_constraints_are_part_of_problem_definition: true,
 Reynolds_number_alone_does_not_uniquely_determine_turbulence_transition: true,
 inviscid_Euler_and_viscous_Navier_Stokes_models_are_distinct: true,
 3D_global_smoothness_for_general_data_remains_open: true,
 numerical_CFD_results_require_grid_time_step_conservation_and_convergence_validation: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 conservation_relation: 0.99,
 incompressibility_relation: 0.99,
 Reynolds_vorticity_relation: 0.99,
 energy_relation: 0.99,
 mathematical_analysis_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(fluid_domain_velocity_pressure_density_viscosity_and_forcing)
 -> classify(incompressible_compressible_steady_or_unsteady_regime)
 -> impose(mass_conservation_and_constitutive_relation)
 -> construct(momentum_equation)
 -> identify(initial_and_boundary_conditions)
 -> nondimensionalise_and_compute(Reynolds_number_if_relevant)
 -> analyse(vorticity_energy_and_flow_regime)
 -> select(exact_asymptotic_weak_or_numerical_solution_method)
 -> enforce(pressure_velocity_coupling_and_incompressibility_if_required)
 -> validate(model_assumptions_regularities_and_numerical_convergence)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed Navier-Stokes semantic graph",
 primary_entity: NAVIER_STOKES_EQUATION,
 preserve: [
  velocity_field,
  pressure_field,
  density,
  viscosity,
  domain,
  forcing,
  initial_conditions,
  boundary_conditions,
  compressibility
 ],
 infer_through: [
  continuity_equations,
  material_derivatives,
  convective_terms,
  viscous stresses,
  Reynolds_numbers,
  vorticity,
  energy_dissipation,
  pressure_Poisson_equations,
  CFD_methods
 ],
 avoid: [
  assuming_divergence_free_velocity_for compressible flow,
  omitting_pressure_constraint_in incompressible flow,
  treating_high_Re_as automatic proof_of turbulence,
  dropping_viscosity_without scale analysis,
  claiming_general_3D_smooth_solution_theory_is settled,
  trusting_CFD_without grid_and_time convergence analysis
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(Navier_Stokes_context)
 -> compare(existing_relations)
 -> preserve(fluid_velocity_pressure_density_viscosity_boundary_and_conservation_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + NAVIER_STOKES_EQUATION
 + VELOCITY_FIELD
 + PRESSURE_FIELD
 + DENSITY
 + VISCOSITY
 + MOMENTUM_CONSERVATION
 + MASS_CONSERVATION
 + CONTINUITY_EQUATION
 + INCOMPRESSIBILITY
 + REYNOLDS_NUMBER
 + VORTICITY
 + TURBULENCE
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> NAVIER_STOKES_EQUATION
 -> FLUID_PROBLEM_IDENTIFICATION
 -> FLOW_REGIME_CLASSIFICATION
 -> CONSERVATION_CONSTITUTIVE_ANALYSIS
 -> MOMENTUM_EQUATION_CONSTRUCTION
 -> CONDITION_ANALYSIS
 -> NONDIMENSIONAL_REYNOLDS_ANALYSIS
 -> VORTICITY_ENERGY_ANALYSIS
 -> SOLUTION_METHOD_SELECTION
 -> PRESSURE_VELOCITY_ANALYSIS
 -> MODEL_NUMERICAL_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW