Fluid Dynamics Analysis Ontology

flow "FluidDynamicsAnalysisOntology" v1.0

# FLUID DYNAMICS ANALYSIS ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/scientific-applications/fluid-dynamics-analysis-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Scientific Applications"
ENTITY := "Fluid Dynamics Analysis"

FLUID_DYNAMICS_ANALYSIS := {
 type: AppliedContinuumAnalysisFramework,
 domain: ScientificApplications,
 purpose: "Analyse fluid motion using conservation laws, vector calculus, differential equations, dimensional analysis, stability theory and computational methods."
}

CONCEPTS := {
 FLUID_DYNAMICS,
 VELOCITY_FIELD,
 PRESSURE,
 DENSITY,
 VISCOSITY,
 CONTINUITY_EQUATION,
 MOMENTUM_EQUATION,
 NAVIER_STOKES_EQUATION,
 EULER_EQUATION,
 VORTICITY,
 STREAMLINE,
 REYNOLDS_NUMBER,
 BOUNDARY_LAYER,
 LAMINAR_FLOW,
 TURBULENCE,
 DRAG,
 LIFT,
 CFD
}

# ---------- FLUID FIELD ----------
fluid state:

u(x,t)
 -> VELOCITY_FIELD

p(x,t)
 -> PRESSURE_FIELD

rho(x,t)
 -> DENSITY_FIELD

T(x,t)
 -> TEMPERATURE_FIELD
    when thermal effects included

FLUID_DYNAMICS
 -> continuum_field_description

# ---------- MATERIAL DERIVATIVE ----------
for scalar or vector field
 phi:

Dphi/Dt
 :=
partial_phi/partial_t
+
u dot grad(phi)

MATERIAL_DERIVATIVE
 -> rate_following(FLUID_PARTICLE)

# ---------- MASS CONSERVATION ----------
general continuity equation:

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

CONTINUITY_EQUATION
 -> local_mass_conservation

# ---------- INCOMPRESSIBLE FLOW ----------
if density constant along motion:

div(u)
 =
0

INCOMPRESSIBILITY
 -> divergence_free_velocity

INCOMPRESSIBLE_FLOW
 -/> necessarily(IRROTATIONAL_FLOW)

# ---------- MOMENTUM ----------
Cauchy momentum equation:

rho*Du/Dt
 =
div(sigma)
+
rho*f

sigma
 -> STRESS_TENSOR

f
 -> BODY_FORCE_PER_UNIT_MASS

MOMENTUM_BALANCE
 -> Newton_second_law_for_continuum

# ---------- NEWTONIAN FLUID ----------
stress:

sigma
 =
-p*I
+
tau

for incompressible constant-viscosity Newtonian fluid:

tau
 =
mu
*
(
 grad(u)
 +
grad(u)^T
)

NAVIER_STOKES
 -> follows_from(MOMENTUM_BALANCE_PLUS_NEWTONIAN_CONSTITUTIVE_LAW)

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

with

div(u)=0

for standard incompressible constant-property form

# ---------- EULER EQUATION ----------
if

mu=0

then

rho
*
Du/Dt
 =
-grad(p)
+
rho*f

EULER_EQUATION
 -> inviscid_flow_model

# ---------- STOKES FLOW ----------
if inertial terms negligible:

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

with

div(u)=0

STOKES_FLOW
 -> low_Reynolds_number_limit

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

=
U*L/nu

REYNOLDS_NUMBER
 -> inertial_to_viscous_ratio

low Re
 -> viscous_effects_dominant

high Re
 -> inertial_effects_dominant

TRANSITION
 -> depends_on(GEOMETRY_DISTURBANCES_AND_BOUNDARIES)

# ---------- STREAMLINE ----------
at fixed time:

dr/ds
 parallel_to
u(r,t)

STREAMLINE
 -> tangent_to(INSTANTANEOUS_VELOCITY_FIELD)

PATHLINE
 -> trajectory_of(INDIVIDUAL_FLUID_PARTICLE)

STEADY_FLOW
 -> streamlines_coincide_with_pathlines
    under standard conditions

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

VORTICITY
 -> local_fluid_rotation_measure

IRROTATIONAL_FLOW
 if

omega
 =
0

# ---------- CIRCULATION ----------
Gamma
 :=
closed_line_integral(
 u dot dl
)

STOKES_THEOREM:

Gamma
 =
surface_integral(
 omega dot n
 dS
)

CIRCULATION
 -> integrated_vorticity_flux

# ---------- VORTICITY EQUATION ----------
for incompressible constant-property 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
 -> major_3D_vorticity_mechanism

# ---------- BERNOULLI ----------
for steady incompressible inviscid flow
along streamline
with conservative body force:

p/rho
+
|u|^2/2
+
g*z
 =
constant

BERNOULLI_EQUATION
 -> mechanical_energy_relation

additional conditions
 -> required_for_global_constant_across_streamlines

# ---------- POTENTIAL FLOW ----------
if

curl(u)=0

then locally:

u
 =
grad(phi)

if also incompressible:

LAPLACIAN(phi)
 =
0

POTENTIAL_FLOW
 -> Laplace_equation_for(VELOCITY_POTENTIAL)

# ---------- STREAM FUNCTION ----------
for 2D incompressible flow:

u
 =
partial_psi/partial_y

v
 =
-
partial_psi/partial_x

STREAM_FUNCTION
 -> automatically_satisfies(CONTINUITY)

streamlines
 -> psi_constant

# ---------- BOUNDARY CONDITIONS ----------
solid wall:

NO_PENETRATION:
 u dot n
 =
u_wall dot n

viscous no-slip:

u
 =
u_wall

free surface:
 -> kinematic_and_dynamic_conditions

BOUNDARY_CONDITIONS
 -> determine(FLOW_SOLUTION)

# ---------- BOUNDARY LAYER ----------
high Reynolds number near wall:

viscous effects
 -> concentrated_in(THIN_REGION)

BOUNDARY_LAYER
 -> strong_velocity_gradient

outside:
 -> approximately_inviscid_flow

inside:
 -> viscosity_essential

# ---------- BOUNDARY LAYER THICKNESS ----------
flat plate scaling:

delta
 approx
L/sqrt(Re_L)

for laminar order-of-magnitude context

BOUNDARY_LAYER_SCALING
 -> derived_from(INERTIA_VISCOSITY_BALANCE)

# ---------- SEPARATION ----------
adverse pressure gradient
 -> slows(NEAR_WALL_FLOW)

wall shear
 -> may_approach_zero

then reverse flow
 -> may develop

BOUNDARY_LAYER_SEPARATION
 -> changes(DRAG_AND_WAKE_STRUCTURE)

# ---------- LAMINAR FLOW ----------
LAMINAR_FLOW
 -> organised_smooth_motion

examples:
 COUETTE_FLOW
 POISEUILLE_FLOW

LAMINAR_SOLUTION
 -> may_be_exact_for(SIMPLE_GEOMETRIES)

# ---------- POISEUILLE FLOW ----------
steady fully developed pipe flow:

pressure gradient
 -> balanced_by(VISCOUS_STRESS)

velocity profile
 -> parabolic

volume flow rate
 -> proportional_to(PRESSURE_DROP)

HAGEN_POISEUILLE_LAW
 -> laminar_circular_pipe_relation

# ---------- TURBULENCE ----------
TURBULENCE
 -> irregular_multiscale_flow

features:
 VORTICES
 FLUCTUATIONS
 MIXING
 ENERGY_CASCADE
 HIGH_EFFECTIVE_TRANSPORT

TURBULENT_FLOW
 -> statistical_analysis_often_required

# ---------- REYNOLDS DECOMPOSITION ----------
u
 =
U_bar
+
u_prime

p
 =
P_bar
+
p_prime

average Navier-Stokes
 -> RANS_EQUATIONS

new term:

-Reynolds_stress
 -> turbulent_momentum_transport

CLOSURE_PROBLEM
 -> requires(TURBULENCE_MODEL)

# ---------- TURBULENCE MODELS ----------
RANS
 -> time_or_ensemble_averaged_model

LES
 -> resolves(LARGE_EDDIES)
 -> models(SMALL_SCALES)

DNS
 -> resolves(ALL_RELEVANT_SCALES)

COMPUTATIONAL_COST:
 DNS > LES > typical_RANS

MODEL_FIDELITY_AND_COST
 -> tradeoff

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

large scales
 -> receive(ENERGY)

nonlinear interactions
 -> transfer_to(SMALLER_SCALES)

viscosity
 -> dissipates_at(SMALL_SCALES)

KOLMOGOROV_SCALE
 -> smallest_classical_turbulent_dissipation_scale
    under standard theory

# ---------- DRAG ----------
fluid force component
 parallel_to(freestream)

D
 :=
1/2
*
rho
*
U^2
*
C_D
*
A

C_D
 -> DRAG_COEFFICIENT

DRAG
 -> pressure_drag
 + skin_friction_drag

# ---------- LIFT ----------
force component
 perpendicular_to(freestream)

L
 :=
1/2
*
rho
*
U^2
*
C_L
*
A

C_L
 -> LIFT_COEFFICIENT

LIFT
 -> depends_on(FLOW_GEOMETRY_CIRCULATION_AND_PRESSURE_DISTRIBUTION)

# ---------- DIMENSIONAL ANALYSIS ----------
Buckingham Pi
 -> construct(DIMENSIONLESS_GROUPS)

important:
 REYNOLDS_NUMBER
 MACH_NUMBER
 FROUDE_NUMBER
 WEBER_NUMBER
 PRANDTL_NUMBER

DIMENSIONLESS_NUMBER
 -> compares(PHYSICAL_EFFECTS)

# ---------- MACH NUMBER ----------
Ma
 :=
U/c_sound

MACH_NUMBER
 -> flow_speed_to_sound_speed_ratio

Ma small
 -> incompressibility_may_be_reasonable
    depending_on(thermal_and_density_context)

COMPRESSIBILITY
 -> important_at(HIGHER_MACH)

# ---------- COMPRESSIBLE FLOW ----------
unknowns:
 rho
 u
 p
 T

equations:
 MASS
 MOMENTUM
 ENERGY
 EQUATION_OF_STATE

COMPRESSIBLE_FLOW
 -> supports(SHOCKS_AND_EXPANSION_WAVES)

# ---------- SHOCK ----------
SHOCK
 -> discontinuity_or_thin_transition_in(FLOW_VARIABLES)

RANKINE_HUGONIOT_CONDITIONS
 -> conservation_across(SHOCK)

ENTROPY_CONDITION
 -> selects(PHYSICALLY_ADMISSIBLE_SOLUTION)

# ---------- FLOW STABILITY ----------
base flow:

U(x)

perturb:

u
 =
U
+
epsilon*u_prime

linearise equations

-> perturbation_evolution

LINEAR_STABILITY_ANALYSIS
 -> determines(GROWTH_OR_DECAY_OF_SMALL_DISTURBANCES)

# ---------- HYDRODYNAMIC INSTABILITY ----------
examples:
 KELVIN_HELMHOLTZ
 RAYLEIGH_BENARD
 TAYLOR_COUETTE
 POISEUILLE_INSTABILITY

INSTABILITY
 -> may_trigger(TRANSITION_OR_PATTERN_FORMATION)

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

E
 =
1/2
*
rho
*
integral(
 |u|^2
 dV
)

viscosity
 -> dissipates(KINETIC_ENERGY)

forcing_and_boundaries
 -> may_supply(ENERGY)

ENERGY_BALANCE
 -> stability_and_turbulence_tool

# ---------- CFD ----------
continuous equations
 -> discretise

methods:
 FINITE_VOLUME
 FINITE_DIFFERENCE
 FINITE_ELEMENT
 SPECTRAL_METHOD

CFD
 -> numerical_solution_of(FLUID_EQUATIONS)

# ---------- FINITE VOLUME ----------
integrate conservation law over
 control volume

fluxes across faces
 -> update(CELL_AVERAGE)

FINITE_VOLUME
 -> local_conservation

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

pressure
 -> enforce(div(u)=0)

methods:
 PROJECTION
 SIMPLE
 PISO
 PRESSURE_CORRECTION

PRESSURE_POISSON
 -> common_constraint_equation

# ---------- MESH ----------
spatial discretisation
 -> MESH

resolution
 -> must_capture(
 BOUNDARY_LAYERS,
 SHEAR_LAYERS,
 WAKES,
 TURBULENT_SCALES
)

GRID_REFINEMENT
 -> test(NUMERICAL_CONVERGENCE)

# ---------- TIME STEP ----------
explicit simulation:

Delta_t
 -> constrained_by(CFL_CONDITION)

CFL
 :=
U*Delta_t/Delta_x

TIME_STEP
 -> controls(NUMERICAL_STABILITY_AND_TEMPORAL_ACCURACY)

# ---------- VALIDATION ----------
CFD result
 -> compare_with(
 ANALYTIC_SOLUTION,
 BENCHMARK,
 EXPERIMENT
)

VERIFICATION
 -> numerical_equation_solution_accuracy

VALIDATION
 -> physical_model_accuracy

FLUID_ANALYSIS
 -> requires(BOTH_WHERE_APPLICABLE)

# ---------- APPLICATIONS ----------
FLUID_DYNAMICS_ANALYSIS
 -> applied_to(AERODYNAMICS)
 -> applied_to(HYDRODYNAMICS)
 -> applied_to(WEATHER)
 -> applied_to(OCEANOGRAPHY)
 -> applied_to(PIPE_FLOW)
 -> applied_to(TURBOMACHINERY)
 -> applied_to(BLOOD_FLOW)
 -> applied_to(COMBUSTION)
 -> applied_to(ENVIRONMENTAL_FLOW)

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

MOMENTUM_CONSERVATION
 -> MOMENTUM_EQUATION

NEWTONIAN_CONSTITUTIVE_LAW
 -> NAVIER_STOKES_EQUATION

VELOCITY_FIELD
 -> VORTICITY

REYNOLDS_NUMBER
 -> compares(INERTIA_AND_VISCOSITY)

BOUNDARY_LAYER
 -> links(VISCOSITY_AND_WALL_FLOW)

TURBULENCE
 -> creates(MULTISCALE_TRANSPORT)

CFD
 -> approximates(FLUID_EQUATIONS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "fluid dynamics analysis" ~= "mathematical analysis of fluid motion using conservation laws and field equations",
 "continuity equation" ~= "local conservation law for fluid mass",
 "material derivative" ~= "rate of change observed while following a fluid particle",
 "vorticity" ~= "curl of the velocity field",
 "Reynolds number" ~= "dimensionless ratio of inertial to viscous effects",
 "boundary layer" ~= "thin near-wall region where viscous velocity gradients are significant"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 incompressible_irrotational_and_inviscid_are_distinct_assumptions: true,
 Bernoulli_equation_requires_specific_flow_assumptions: true,
 Reynolds_number_alone_does_not_uniquely_predict_transition: true,
 turbulence_models_are_approximations_with_model_form_uncertainty: true,
 boundary_conditions_are_part_of_fluid_problem_definition: true,
 compressible_flow_requires_mass_momentum_energy_and_thermodynamic_closure: true,
 CFD_results_require_mesh_time_step_verification_and_physical_validation: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 conservation_relation: 0.99,
 Navier_Stokes_relation: 0.99,
 vorticity_relation: 0.99,
 boundary_layer_relation: 0.99,
 turbulence_CFD_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(fluid_geometry_properties_velocity_pressure_and_forcing)
 -> classify(incompressible_compressible_viscous_inviscid_steady_or_unsteady_flow)
 -> apply(mass_momentum_and_energy_conservation_as_required)
 -> nondimensionalise_and_compute(relevant_dimensionless_numbers)
 -> analyse(vorticity_streamlines_boundary_layers_and_flow_regime)
 -> choose(exact_asymptotic_stability_or_turbulence_model)
 -> select(CFD_discretisation_if_numerical_solution_required)
 -> resolve(pressure_velocity_boundary_and_initial_conditions)
 -> verify(mesh_time_step_conservation_and_solver_convergence)
 -> validate_against(experiment_benchmark_or_reference)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed fluid-dynamics-analysis semantic graph",
 primary_entity: FLUID_DYNAMICS_ANALYSIS,
 preserve: [
  geometry,
  fluid_properties,
  velocity,
  pressure,
  density,
  viscosity,
  forcing,
  boundary_conditions,
  initial_conditions,
  flow_regime
 ],
 infer_through: [
  continuity,
  momentum_balance,
  Navier_Stokes,
  Euler_flow,
  vorticity,
  Reynolds_number,
  boundary_layers,
  turbulence,
  CFD
 ],
 avoid: [
  conflating_incompressible_with irrotational,
  applying_Bernoulli_without assumptions,
  predicting_turbulence_from Reynolds_number alone,
  ignoring_wall_boundary_conditions,
  treating_turbulence_model_as exact physics,
  accepting_CFD_without convergence_and validation
 ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + FLUID_DYNAMICS_ANALYSIS
 + VELOCITY_FIELD
 + PRESSURE
 + DENSITY
 + VISCOSITY
 + CONTINUITY_EQUATION
 + MOMENTUM_EQUATION
 + NAVIER_STOKES_EQUATION
 + VORTICITY
 + REYNOLDS_NUMBER
 + BOUNDARY_LAYER
 + TURBULENCE
 + CFD
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> FLUID_DYNAMICS_ANALYSIS
 -> FLOW_PROBLEM_IDENTIFICATION
 -> REGIME_CLASSIFICATION
 -> CONSERVATION_ANALYSIS
 -> DIMENSIONLESS_ANALYSIS
 -> FLOW_STRUCTURE_ANALYSIS
 -> MODEL_METHOD_SELECTION
 -> CFD_DISCRETISATION
 -> CONDITION_COUPLING_ANALYSIS
 -> NUMERICAL_VERIFICATION
 -> PHYSICAL_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW