compute: cpu

Gas

Gas
Thermo
γ
R, J/(kg·K)
γ = 1.400 · R = 287.05 J/(kg·K)calorically perfect gascp(T) applies to Nozzle / Euler

Reservoir & back pressure

Stagnation pressure p₀
bar(a)
Stagnation temperature T₀
K
Back pressure pᵦ/p₀0.700
Back pressure pᵦ3.500 bar(a)

Nozzle geometry

Throat area A*
cm²
Exit ratio Aₑ/A*2.50
Nozzle length L
m
Drag the upper or lower wall. Control points remain monotonic so the model keeps one throat and one divergent section.

Coupled viscous nozzle

Viscous coupling
Boundary-layer state
Transition Reₛ
Thermal wall
Wall temperature Twall
K
Wall heat flux q″
kW/m², + gas→wall
Wall material
Wall thickness
mm
Wall k @ 300 K
W/(m·K)
dk/dT
W/(m·K²)
Coolant
Coolant inlet T
K
Coolant inlet p
bar(a)
Coolant mass flow
kg/s
Coolant direction
Coolant geometry
Number of channels
Channel height
mm
Rib width
mm
Wall roughness ε
mm
Channel width follows the local circumferential pitch: w(x)=2πr/N−rib. Height and rib width stay constant; flow area, Dₕ and cooled coverage therefore vary with nozzle radius.
Total flow area
mm²
Hydraulic diameter
mm
Cooled coverage
Thermal restraint χ
Wall E @ 300 K
GPa
dE/dT
GPa/K
Wall α @ 300 K
µm/(m·K)
dα/dT
µm/(m·K²)
Poisson ν
Yield @ 300 K
MPa
dσᵧ/dT
MPa/K
Channel-roof Cᵦ
Roof-bending strip proxy: σᵣₒₒ𝒻=Cᵦ·|Δp|·(w/t)². Cᵦ=0.50 corresponds to a fixed-fixed strip idealization; use as a screening coefficient, not a plate/FEA substitute.
Cyclic K′
MPa
Cyclic n′
Neuber hotspot Kₜ
Local cyclic screen: ε=σ/E+(σ/K′)^(1/n′), with Neuber energy density σ·ε=(Kₜσelastic)²/E. Kₜ=1 adds no extra notch concentration beyond the reduced-order roof/membrane/thermal screen.
LCF strain multiplier Kε
Fatigue σ′f
MPa
Fatigue b
Fatigue ε′f
Fatigue c
Duty A cycles
Duty A severity
Duty B cycles
Duty B severity
Duty C cycles
Duty C severity
Coffin–Manson–Basquin uses the Neuber/Ramberg–Osgood local strain range. Duty A/B/C severity multiplies the whole static screening load/range uniformly; damage is summed linearly with Miner. The directed-plastic-strain value is a static spectrum susceptibility proxy; the explicit transient below separately integrates local kinematic plasticity and optional creep.
Explicit hot/cold duty-cycle transient
Transient screen
Wall density ρ
kg/m³
Wall cp
J/(kg·K)
Startup ramp
s
Hot dwell
s
Shutdown ramp
s
Cold dwell
s
Cycles simulated
Steps / cycle
Cold-soak T
K
Off-state gas p
bar(a)
Kinematic Hₖ
GPa @ 300 K
Creep rate @ ref.
%/h
Creep σref
MPa
Creep exponent n
Creep Tref
K
Creep Q/R
K
Local transient post-processor: steady hot-state Taw/hg, coolant h/p and pressure loads are the firing endpoint. The firing command uses smooth ramps; coolant flow and local pressure are assumed maintained through shutdown. A two-node implicit wall RC model supplies thermal lag. Incremental plasticity uses a signed uniaxial equivalent path with linear kinematic hardening. Creep rate=0 disables the Norton–Arrhenius hot-dwell term.
Coolant cp
J/(kg·K)
Coolant density ρ
kg/m³
Coolant viscosity μ
Pa·s
Coolant conductivity k
W/(m·K)
Reduced-order channel model. Water mode uses pressure-dependent IAPWS-IF97 Region-1 thermodynamics with IAPWS viscosity and conductivity base terms, plus the Region-4 saturation screen. Wall mechanics adds membrane, channel-roof strip bending, temperature-dependent elastic properties, Neuber/Ramberg–Osgood cyclic plasticity screening, Miner duty-spectrum damage and a directed-plastic ratcheting proxy; none replace structural FEA, shakedown/creep analysis or qualified fatigue data.
Overexpansion check
pwall/pamb threshold
Axisymmetric circular equivalent. Integral boundary layer is marched along the wall: Thwaites-type laminar closure, Head-type turbulent entrainment, pressure-gradient history and auto transition. Thermal wall modes use recovery temperature plus a Colburn/Reynolds analogy; heat transfer changes local h₀ while wall shear generates entropy, so Aeff ↔ quasi-1D core ↔ heat transfer are iterated together. Reduced-order model, not RANS or SWBLI CFD. Conjugate mode adds 1D wall conduction and bulk coolant heating/pressure drop; it is not a resolved solid/fluid conjugate CFD solution.

Display

Colour field
Particles
Post-exit shock cells are qualitative geometry cues. Values and charts inside the nozzle are the quasi-1D solution. In cp(T) mode, Air/N₂/CO₂ properties use NIST Shomate fits; endpoint cp is extrapolated outside the tabulated temperature range and flagged in the statistics.
Laval nozzle——BL off
Move the cursor over the nozzle to link all charts. Drag a wall to edit the area profile.
Mach number M(x)
Static pressure p(x)
Temperature T(x)
Density ρ(x)
Velocity u(x) and sound speed a(x)
Total pressure p₀(x)

Nozzle operating map

Background curves are inviscid references. With viscous coupling enabled, the current choking threshold, shock-at-exit threshold and design pressure are re-solved on Aeff.

Mass flow vs pressure ratio

T–s / irreversibility

—

Coupled boundary layer

Enable viscous coupling to iterate displacement blockage against the quasi-1D core, include wall-friction p₀ loss, and re-solve choking and shock position.

Integral BL thickness

θ and δ* are marched with pressure-gradient history; H and β are available in the cursor and colour field.

Wall heat transfer

Adiabatic by default. Prescribed Twall, q″ or conjugate coolant couples wall heat transfer back into local stagnation enthalpy and the nozzle solution.

Conjugate cooling

Select Conjugate coolant to solve hot-wall, cold-wall and bulk-coolant temperatures together with channel pressure drop.

Boiling / subcooling margin

Water mode compares local coolant and cold-wall temperatures with IAPWS saturation temperature. This is an onset-of-boiling / single-phase validity screen, not a CHF correlation.

Wall stress screen

Reduced-order membrane + channel-roof strip bending + restrained thermal stress screen. It does not replace plate/solid mechanics or structural FEA.

LCF / strain-life screen

Coffin–Manson–Basquin screening from the local Neuber / Ramberg–Osgood cyclic strain range. User-supplied constants are required for material-specific interpretation.

Cyclic plasticity / ratcheting

Static Neuber + cyclic Ramberg–Osgood strain and a directed-plastic-strain proxy over the A/B/C duty spectrum. The explicit hot/cold transient panels below separately integrate linear-kinematic plasticity and optional dwell creep.

Axial transient risk map

Runs the hot/cold transient independently at every eligible nozzle station and compares where thermal, fatigue, residual-strain and creep demand control.

Hot/cold wall transient

Two-node wall thermal inertia at the governing axial screening location.

Incremental stress–strain loop

Last simulated cycle of the local signed equivalent hotspot path.
—
—

Area–Mach relation explorer

A/A*2.50
A/A* = (1/M) [ 2/(γ+1) · (1 + (γ−1)M²/2) ](γ+1)/(2(γ−1))
Except at A/A*=1, one area ratio has a subsonic and a supersonic isentropic solution.

Selected state ratios

Branchp/p₀T/T₀ρ/ρ₀

Two branches

Nozzle locations with similar area

Constant-area adiabatic duct + friction

Inlet Mach M₁
Inlet pressure p₁
bar(a)
Inlet temperature T₁
K
Darcy friction factor fᴅ
Length ratio L/D
fᴅL*/D = (1−M²)/(γM²) + (γ+1)/(2γ) ln[(γ+1)M²/(2+(γ−1)M²)]
Subsonic flow accelerates toward M=1; supersonic flow decelerates toward M=1. T₀ stays constant while p₀ falls.
—
—

Mach along duct

Pressure & total pressure

Fanno line in T–s coordinates

Constant-area frictionless duct + heat transfer

Inlet Mach M₁
Inlet pressure p₁
bar(a)
Inlet temperature T₁
K
Heat q
kJ/kg
T₀/T₀* = (γ+1)M²[2+(γ−1)M²] / (1+γM²)²
Positive heat addition drives both branches toward thermal choking at M=1. Negative q moves the state away from M=1 along the same Rayleigh branch.
—
—

Mach and T₀

Static p and T

Rayleigh line in T–s coordinates

Supersonic turning

Upstream Mach M₁
Wall turn θ12.0° compression
Attached-shock branch
Positive θ turns into the flow and produces a compression shock when an attached solution exists. Negative θ produces a centered Prandtl–Meyer expansion fan.
—
—

Wedge / expansion corner

θ–β–M diagram

Unsteady quasi-1D Euler

Scenario
Finite-volume cells
CFL number
Spatial / time scheme
Slope limiter
Boundary treatment
Outlet pressure relaxation
0…1
Smaller relaxation reduces acoustic reflection but enforces the target pressure more slowly; 1 approaches a hard imposed static-pressure boundary. The reservoir inlet uses the outgoing Riemann invariant with specified p₀,T₀.
Replay duration
s
Colour field
Uses the current Laval geometry, p₀, T₀ and pᵦ from the Nozzle tab. The duct starts at rest at pᵦ; the plenum boundary is opened at t=0.
Initial pᵦ/p₀
Final pᵦ/p₀
Initial condition is the analytic quasi-steady nozzle solution at the first pressure ratio; at t=0 the outlet static pressure is stepped to the second.
Tube length L
m
Tube inside diameter
mm
Reservoir p₀
bar(a)
Initial tube p
bar(a)
Gas temperature
K
Capacity input
Full-open capacity
m³/h (Kv)
Equivalent Cd·A—
Opening law
Opening time
ms
The valve is a lumped compressible restriction at the tube inlet. Kv/Cv are converted to an equivalent Cd·A; the gas flow then uses an ideal-gas compressible-orifice relation and automatically chokes when the downstream/upstream pressure ratio falls below the critical value.
Uses the left/right states, diaphragm position and tube length from the Shock tube tab. Numerical finite-volume pressure is overlaid with the exact ideal-gas Riemann solution until the first end-wall interaction.

Time

Physical time0 ms
Snapshots are stored for replay and the x–t heatmap.
READY
Reset to initialize the selected scenario.
Finite-volume fieldHLLC · MUSCL–TVD · SSPRK2characteristic BCreference: —

Pressure profile

Mach profile

x–t pressure history

Each row is a stored numerical snapshot. Shock thickness is numerical, not a physical viscous thickness. MUSCL reduces diffusion but does not make the discontinuity physically thin.

Valve mass flow history

Positive ṁ is reservoir → tube. The curve reflects both commanded opening and the instantaneous pressure ratio across the valve.

Initial states

Driver / left

pᴸ
bar(a)
Tᴸ
K

Driven / right

pᴿ
bar(a)
Tᴿ
K
Tube length L
m
Diaphragm x/L0.50
Time0 ms
The physical time range stops before the first wave reaches an end wall. Replay time is stretched for visibility; the displayed time is the physical time.

Tube state

x–t wave diagram

Pressure profile p(x,t)

Built-in analytic self-checks

These are implementation checks, not independent validation.

Parameter studies

Each study is an ensemble of independent solves on the shared compute runtime. Every case is a full solve of the same model this page shows — nothing is reduced or surrogated. With no runtime and no Worker the same cases run one after another on this thread, which is slow but identical.
idle