Two-Fluid Turbulence Model Tracks Supersonic Jet With Striking Accuracy

Simulating a supersonic jet is one of the hardest problems in computational fluid dynamics. At high Mach numbers, turbulence becomes strongly anisotropic, shock waves interact with swirling eddies, and compressibility suppresses the mixing that conventional models struggle to capture. A team at an aerospace research group now reports that an old idea, reworked into a modern mathematical framework, can meet this challenge: a two-fluid turbulence model that treats turbulence as two interpenetrating fluids sliding past each other, validated against one of the most carefully measured jet experiments ever conducted.

The study, published as open access research in Discover Mechanical Engineering, presents numerical simulations of an axisymmetric jet exhausting from a Laval nozzle at a Mach number of 1.63 under temperature-matched conditions, meaning the static temperature of the exhaust gas equals that of the ambient air. This configuration eliminates mean thermal gradients, so the turbulent mixing can be studied in its purest form. Because the case allows highly accurate measurements of density, pressure, and velocity profiles, it is included in the well-known NASA experimental database and serves as a standard benchmark for verifying turbulence models.

The model’s lineage is unusual. The two-fluid concept was first proposed by Spalding in 1983, representing turbulence as a mixture of two fluids in relative motion. Numerous studies demonstrated its predictive potential, yet it never found wide practical adoption. The obstacle was mathematical: the turbulent quantities required additional transport equations built on empirical hypotheses, leaving a formulation that resembled conventional Reynolds-averaged Navier–Stokes models but with roughly twice the governing equations. Later work by one of the present authors restructured the approach, introducing mean and relative flow variables that decompose the governing equations into separate systems for the mean and relative flows.

This restructuring is the heart of the new model. In the mean-flow equations, the Reynolds stresses appear naturally as products of the relative velocity components, rather than being imposed through an eddy-viscosity assumption. The relative-velocity equations contain interaction forces between the two fluids proportional to the relative velocity, including a Saffman force term coupling the mean rotation to the relative motion. These interaction forces are approximated empirically as functions of the mean and relative velocities, closing the system without extra transport equations. The authors identify this closure as the principal distinction between their model and conventional RANS approaches based on the k–ε, k–ω, or SST families.

The physical setup is precisely defined. The nozzle exit diameter is 50.8 millimeters, and the nozzle contour was designed by the method of characteristics to produce an almost shock-free diverging flow with nearly perfectly expanded exit conditions. The governing parameters are stated explicitly: the ratio of stagnation to reference temperature is 1.53, the jet Mach number is 1.63, the Reynolds number based on jet velocity and exit radius is 5768, the stagnation pressure is 4.44 times the ambient reference pressure, and the jet exit velocity is 555 meters per second. The computational domain begins at the nozzle exit plane, so nozzle-wall effects on the downstream flow are neglected, and the friction and heat-transfer coefficients simplify accordingly.

The numerical machinery reflects decades of accumulated practice. The governing equations, formulated in cylindrical coordinates and simplified with Prandtl’s boundary-layer theory, were solved with an in-house code using a pseudo-time marching approach to reach steady state. A coordinate transformation refines the mesh near the axis of symmetry, where flow gradients are strongest, and a staggered grid arrangement enforces pressure–velocity coupling through the SIMPLE algorithm. Convective fluxes use a second-order upwind scheme with flux limiters to suppress spurious oscillations, diffusion terms use implicit central differences, and non-reflecting boundary conditions at the outlet and far field prevent spurious pressure reflections. Grid-convergence tests with transverse spacings of 0.01 and 0.02 showed satisfactory convergence at the finer spacing.

The validation results are the paper’s centerpiece. Along the jet centerline, the nondimensional axial velocity predicted by the two-fluid model tracks the experimental data substantially more accurately than the SST-VM model, the only numerical results provided in the NASA validation database for this configuration. Axial velocity profiles at cross-sections from x/D = 2 to 16, covering the jet core decay and the growth of the mixing layer, also agree closely with measurements, with tabulated relative errors favoring the two-fluid model. Radial distributions of turbulent stress at the same stations show similarly good agreement, a demanding test because turbulent stresses are the quantities most sensitive to anisotropy and compressibility effects.

An additional comparison exploits the model’s physical interpretation. In the two-fluid formulation, the relative velocity components essentially represent root-mean-square values of the corresponding velocity fluctuations. The computed normalized RMS velocity profiles at different axial stations are in satisfactory agreement with the experimental data. For this quantity, no SST-VM results exist in the NASA database, so the comparison there rests on the experiment alone. Contours of the flow velocity magnitude complete the picture of the predicted jet structure.

The authors conclude that the model achieves an effective balance between computational efficiency and predictive accuracy, maintaining substantially lower computational costs than high-fidelity approaches such as large eddy simulation and hybrid LES/RANS methods, whose expense limits their use at the large Reynolds numbers typical of practical aerospace problems. They position the method for aerospace applications, nozzle gas-dynamic calculations, jet propulsion systems, and aeroacoustic prediction, and describe the results as a foundation for extending turbulence modeling toward anisotropy, shock–turbulence interactions, and thermodynamic nonequilibrium effects.

Some caveats belong in any careful reading. This is a single benchmark case, albeit a demanding and well-characterized one, simulated with the authors’ own code and empirical constants; broader validation across other Mach numbers, heated jets, and shock-containing flows remains future work. The reported agreement is the authors’ own comparison against published experimental data, and the claimed advantages over SST-VM rest on the tabulated relative errors for this configuration. Still, for a flow regime where eddy-viscosity models have long fallen short, the demonstration that a closed two-fluid system, without a thicket of extra transport equations, can reproduce jet core decay, mixing layer growth, and turbulence suppression at high Mach number is a result worth the attention of anyone modeling compressible turbulent flows.

Subject of Research: Numerical simulation of a supersonic axisymmetric jet using a two-fluid turbulence model validated against NASA experimental data.

Article Title: Numerical investigation of supersonic jet using a two-fluid turbulence model

Article References: Malikov, Z., Nazarov, F., Amirova, D., & Omonov, Z. (2026). Numerical investigation of supersonic jet using a two-fluid turbulence model. Discover Mechanical Engineering, 5(1), Article 195. https://doi.org/10.1007/s44245-026-00359-9

Image Credits: AI Generated

DOI: 10.1007/s44245-026-00359-9

Keywords: supersonic jet, turbulence modeling, two-fluid model, computational fluid dynamics, Reynolds stress anisotropy, compressibility effects, NASA benchmark, RANS, SST-VM, aerospace engineering, jet noise, Laval nozzle