Pressure mismatch · compressible flow

Supersonic Jet & Shock Diamonds

Why does a supersonic jet form a chain of diamonds when its exit pressure differs from the surrounding air? A jet that leaves the nozzle above the surrounding pressure expands first; one that leaves below it is squeezed first. Each overshoot reflects off the jet's edge as the opposite kind of wave, so expansions and compressions alternate down the jet and draw the diamond pattern. Matched to the surrounding pressure, an ideal jet with no viscosity stays straight.

A converging–diverging nozzle feeds a planar jet that the Euler equations compute as you watch. Compare underexpanded, matched and overexpanded jets in the Mach, pressure and numerical schlieren views. Follow the waves along the centerline, check that the open box conserves mass, momentum and energy, and test the code on a normal shock whose exact jump is known.

Simulation time

t

$0$

Nozzle total pressure

p_0/p_a

$0$

Exit / throat area

A_e/A_*

$0$

Inlet mass flow

\dot m

$0$

Ideal nozzle thrust

F

$0$

Probe Mach

M_p

$0$

Probe pressure

p_p/p_a

$0$

Probe pressure in kPa

p_p\;[\mathrm{kPa}]

$0$

Open-box conservation error

\varepsilon_U

$0$

Relative state-change rate

r_U

$0$

Normal-shock pressure error

\varepsilon_s

$\mathrm{n/a}$

First-order fallback steps

n_{\mathrm{lim}}

$0$

Step cost

\mathrm{ms}

$0$

Nozzle flow and the plume it makes

Match the exit pressure to the surrounding air

Jet mass-fraction contourA passively transported contact layer, not a solid wall.Compression hintsPressure-gradient cells with negative velocity divergence; not an exact shock locator.Exit / upstream Mach

M_e

2

Exit / ambient pressure

p_e/p_a

1.8

Specific-heat ratio

\gamma

1.4

Quasi-1D nozzle length

L_n

2.5

Quasi-1D contour shape

q_n

1

Ambient pressure in kPa

p_a\;[\mathrm{kPa}]

101.3

Grid resolution

N

1

Multidimensional CFL

\mathrm{CFL}

0.4

Requested simulation pace

s

1

Field layer

\mathcal V

2

Click the flow field or the centerline chart to move the probe; once the canvas has focus, the arrow keys move it too. Changing a physical or grid setting restarts the run, but changing the view or the pace does not. After you edit a preset, the actual pressure ratio still decides whether the jet is underexpanded, matched or overexpanded.

Normalized units: $R=T_0=T_a=p_a=D_e=1$ . The full box outside the nozzle is 12 exit widths long and 5 high. A phone shows only the near field, but the whole box is still computed. The run starts from a straight jet that already fills its path and then relaxes. Give the waves time to develop, because the early images are transients.

The solver uses MUSCL reconstruction, a hybrid HLLC/HLLE flux and SSP-RK2 time stepping. Compressive pressure jumps use HLLE. A step that gives an inadmissible state is retried at first order, then with half the time step. No floor is applied to density or pressure, and the energy is never corrected. The conservation check integrates the actual flux through the boundary; a small error does not prove grid convergence.

Nozzle length and contour shape change the quasi-1D area and Mach number profiles, but not the chosen exit state. Ambient pressure sets the dimensional pressure scale: at a fixed ratio of exit to ambient pressure, the normalized inviscid plume is identical. Mass flow and thrust stay normalized and are not SI predictions. The exit sits in a nozzle plate with a slip wall. The ambient reservoirs and the downstream extrapolation are approximate open boundaries and do not absorb outgoing waves perfectly. At high resolution, the limited computing time per frame can slow the simulation.

The gas is calorically perfect. The model has no combustion, viscosity or heat conduction, and it does not predict sound. The upstream nozzle is a quasi-1D isentropic calculation and is not part of the 2D grid. The jet outside the nozzle is planar, not axisymmetric. The allowed pressure range assumes the flow leaves the nozzle supersonic, with no shock inside the nozzle. Numerical schlieren also shows contact layers, so a dark edge alone does not prove there is a shock.

Physics tutorial

A pressure mismatch creates a system of waves

BackgroundA supersonic nozzle does not automatically exit at the pressure of the surrounding air. Once free, the jet adjusts through expansion and compression waves. As these waves cross the jet and interact, they can form a repeating chain of shock cells.

Why it mattersAn animation of glowing diamonds cannot tell you whether the pressure, mass flow or shock jump makes physical sense. This Lab gives you four things to check separately: the nozzle calculation, the computed flow outside the nozzle, the profile along the centerline and a conservation check for the whole box.

Start with the essentials

Focus question: What changes when exit pressure is above, equal to, or below ambient pressure?
One-sentence intuition: A jet that leaves above ambient pressure first expands, and one that leaves below it first compresses. With matched pressure, a straight inviscid jet is already a steady solution whose edge is a stationary contact and shear layer, so the solver forms no diamonds in that limit.

Core mathematical model

Nozzle area and Mach number

\frac{A}{A_*}=\frac1M\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}},\quad \frac{p}{p_0}=\left(1+\frac{\gamma-1}{2}M^2\right)^{-\frac{\gamma}{\gamma-1}}

Every area larger than the throat has two solutions, one subsonic and one supersonic. The nozzle calculation takes the subsonic one before the throat and the supersonic one after it. Total temperature is fixed, and total pressure follows from the chosen exit state. Shocks inside the nozzle and boundary layers on its walls are not solved.

Maximum mass flow through a sonic throat, and ideal thrust

\dot m=\rho u A=\frac{A_*p_0}{\sqrt{T_0}}\sqrt{\frac\gamma R}\left(\frac2{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}},\quad F=\dot m u_e+(p_e-p_a)A_e

For a fixed area and fixed total pressure and temperature, the mass flow is largest where the flow reaches the speed of sound. Every cross-section of the nozzle carries the same mass flow. The thrust is the ideal quasi-1D nozzle value per unit span; it is not a force measured on the 2D box. The critical pressure ratio describes the local sonic throat. It is not a back-pressure threshold that holds for every converging–diverging nozzle.

Exact jump across a normal shock

\frac{p_2}{p_1}=1+\frac{2\gamma}{\gamma+1}(M_1^2-1),\quad \frac{\rho_2}{\rho_1}=\frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2},\quad M_2^2=\frac{1+\frac{\gamma-1}{2}M_1^2}{\gamma M_1^2-\frac{\gamma-1}{2}}

The fluxes of mass, axial momentum and energy are equal on both sides of the shock, but total pressure drops across it. The normal-shock test runs this stationary jump through the same 2D code, and the dashed centerline curve comes straight from the exact relations. The error is the mean absolute pressure error divided by the downstream pressure. The test checks how the code handles a shock; it does not predict that every jet has a Mach disk.

External two-dimensional Euler flow

\partial_t\mathbf U+\partial_x\mathbf F+\partial_y\mathbf G=0,\quad \mathbf U=(\rho,\rho u,\rho v,\mathcal E,\rho c)^{\mathsf T},\quad p=(\gamma-1)\left[\mathcal E-\tfrac12\rho(u^2+v^2)\right]

The code updates density, both momentum components, energy and a passive jet mass fraction in conservative form. MUSCL reconstruction uses monotonized-central slopes of the primitive variables. The flux is a hybrid of HLLC and HLLE with wave-speed bounds from the Roe and Davis estimates; where the flow compresses and the pressure jumps by more than one tenth of the smaller pressure, HLLE takes over. SSP-RK2 advances the unsplit flux divergence in time.

Time-step limit and conservation in an open box

\Delta t\le\frac{\mathrm{CFL}}{\max\left[(|u|+a)/\Delta x+(|v|+a)/\Delta y\right]},\quad a=\sqrt{\gamma p/\rho},\quad \mathbf Q(t)+\int_0^t\mathbf\Phi_{\partial\Omega}\,dt=\mathbf Q(0)

Mass, momentum and energy can enter or leave the box through its boundaries. The conservation check adds the outward numerical flux, weighted by Runge–Kutta stage, to the amount currently inside the box. Quantities that start at zero are scaled by the initial mass instead. The relative state-change rate is a dimensionless vector norm per unit time; it does not estimate the discretization error. A rejected update is retried first without reconstruction and then with a smaller time step; the code never clips values to repair the energy.

Numerical schlieren shows density gradients

S=1-\exp\left(-0.16|\nabla\rho|\right)

This image takes centered-difference gradients of the same finite-volume density field and shows them with fixed contrast. Shocks and contact layers can both appear dark. A cell gets a compression hint only when it has a pressure gradient and negative velocity divergence. The hints mark grid cells above a threshold; they do not give the exact shock shape and are not a calibrated optical measurement.

Common difficulties

The colors are not flame

Typical misconceptionThe bright regions show the exhaust burning and how hot it is.

Better mental modelThe model has no combustion and no light emission. The colors show Mach number or pressure, and the black-and-white view shows density gradients. A planar inviscid jet also cannot predict the cell spacing of an axisymmetric jet, viscous mixing, real-gas chemistry or noise.

Conservation does not mean grid convergence

Typical misconceptionIf the conservation error is tiny, the shock-cell pattern must be right.

Better mental modelThe conservation check only shows that the discrete fluxes neither create nor lose conserved quantities. Numerical diffusion, reconstruction, shock detection, the box boundaries and the grid resolution still shape the pattern. Run again on different grids and compare at the same simulation time. In the early frames the prefilled jet is still relaxing; that transient is not yet a steady plume.

Run the experiment

01
Mismatch the exit and ambient pressure
Run the three plume presets at the same Mach number. Let the waves develop, then look at the pressure and schlieren views. Click the centerline to move the probe that all views share.
What to observe: The underexpanded jet expands first and the overexpanded jet compresses first. The matched jet stays straight, bounded by a shear and contact layer. The waves come out of the flow solution itself; no sine-shaped diamonds are drawn in.
02
Test the code on a single shock
Select the Normal-shock test. Let it run, then raise the Grid resolution. Compare the pressure curve with the dashed exact jump and read the shock error.
What to observe: The computed shock is spread over several cells. A finer grid should shrink its integrated error. The ratios between the two plateaus and the equal fluxes of mass, momentum and energy are independent checks against the exact theory.
03
Change the nozzle and the grid
Change the Mach number, the specific-heat ratio and the nozzle length. Compare the exit-to-throat area ratio and the mass flow. Then raise the Grid resolution and watch the step cost and the actual simulation time.
What to observe: The nozzle shape changes, but every section still carries the same mass flow. Length does not change the chosen exit state. A finer grid may advance more slowly because each frame has limited computing time. Even a small conservation error still needs a comparison between grids.
04
Scientific basis and inspiration
NASA Glenn: Isentropic Flow Equations, Mass Flow Choking, Normal Shock Wave Equations, and Rocket Thrust Equations at grc.nasa.gov/www/k-12/airplane/. HLLC basis: Toro, Spruce and Speares (1994), Restoration of the contact surface in the HLL-Riemann solver, DOI 10.1007/BF01414629.
What to observe: Bugman123 FluidMotion inspired the idea only; the code and graphics are original. Automated tests cover analytic nozzle and shock values, preservation of a stationary contact, the independent Sod solution, grid refinement, symmetry, positivity and conservation of boundary flux.

Conceptual inspiration: FluidMotion. Original simulations and graphics; the model limits are described above.

Core mathematical model

Nozzle area and Mach number

Maximum mass flow through a sonic throat, and ideal thrust

Exact jump across a normal shock

External two-dimensional Euler flow

Time-step limit and conservation in an open box

Numerical schlieren shows density gradients

Common difficulties

The colors are not flame

Conservation does not mean grid convergence

Run the experiment

Mismatch the exit and ambient pressure

Test the code on a single shock

Change the nozzle and the grid

Scientific basis and inspiration