Reinterpreting shock wave structure predictions using the Navier–Stokes equations (original) (raw)

Recasting Navier-Stokes equations: Shock wave structure description

INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2019, 2020

Classical Navier-Stokes equations are known to be inadequate in describing some flows in both the compressible and incompressible configurations. A ubiquitous simple example of the failure is the description of shock wave structure. A new compressible hydrodynamic set of equations is here developed using a velocity transformation technique similar in nature to Lorentz transformation and termed "re-casted Navier-Stokes equations". We found that results with the re-casted Navier-Stokes equations fit better the experimental data.

A Full Evaluation of Accurate Constitutive Relations for Shock Wave Structures in Monatomic Gases

2021

We present a full investigation into shock wave profile description using hydrodynamics models. We identified constitutive equations that provide better agreement for all parameters involved in testing hydrodynamic equations for the prediction of shock structure in a monatomic gas in the Mach number range 1.0 − 11.0. Compared to previous studies that focussed mainly on the density profile across the shock, here we also include temperature profiles as well as non-negativity of entropy production throughout the shock. The results obtained show an improvement upon those obtained previously in the bi-velocity hydrodynamics and are more accurate than in the hydrodynamic models from expansions method solutions to the Boltzmann equation.

Direct numerical simulation of a Mach 2 shock interacting with isotropic turbulence

Applied scientific research, 1995

Direct Numerical Simulation (DNS) and linear analysis of a shock interacting with incompressible and compressible isotropic turbulence is conducted. A dependence of amplification ratios on the degree of compressibility of the incoming flow is found. It can be shown that the enhancement of rms values of turbulent quantifies across the shock varies according to the ratio of compressible to incompressible kinetic energy X (exact definition see eq. 8). Inflow conditions with high values of X display reduced amplification ratios of TKE and thermodynamic quantities while vorticity fluctuations are enhanced more strongly. The different behaviour of the turbulent kinetic energy (TKE) is due to the reduced pressure diffusion term in the TKE-equation. Experiments show qualitatively a similar behaviour as the simulation with incompressible inflow conditions, but they could so far not confirm our findings of reduced amplification rates in the compressible case, one of the reasons being the lack of knowledge of all flow parameters upstream of the shock front and the inability to generate isotropic turbulence in real life experiments. For the DNS we use a third order in space shock-capturing scheme based on the ENO algorithm of Harten [10] together with an approximate Riemann solver. This non-TVD scheme turned out to have many advantages over other common Godunov-type high resolution schemes for the specific problem of a shock interacting with turbulent fields.

Numerical simulation of shock wave structure in nitrogen

Physics of Fluids, 2007

The one-dimensional problem of the structure of a stationary shock wave in nitrogen is solved in the frame of the Navier-Stokes ͑NS͒ equations. Proper interpretation of the bulk viscosity coefficient included in the shear stress tensor leads to a numerical solution close to the experiment, showing that the NS equations provide more accurate solutions to the problem than supposed previously.

Shock calculations using a very high order accurate Euler and Navier-Stokes solver

2008

Initial attempts and experience with shock calculations using a very high order (2, 4 and 6 th order) stable Euler/Navier-Stokes finite difference solver is discussed. The code is built on numerical techniques developed during the last decade by Uppsala University, Sweden and NASA Langley, USA, see Svärd et al. [4], . The main features of the code are: the finite difference operators on summation-by-parts (SBP) form, the weak implementation of boundary and interface conditions and artificial dissipation operators on SBP form.

The non-linear refraction of shock waves by upstream disturbances in steady supersonic flow

Journal of Fluid Mechanics, 1970

The general problem studied is the propagation of an oblique shock wave through a two-dimensional, steady, non-uniform oncoming flow. A higher-order theory is developed to treat the refraction of the incident oblique shock wave by irrotational or rotational disturbances of arbitrary amplitude provided the flow is supersonic behind the shock. A unique feature of the analysis is the formulation of the flow equations on the downstream side of the shock wave. It is shown that the cumulative effect of the downstream wave interactions on the propagation of the shock wave can be accounted for exactly by a single parameter Φ, the local ratio of the pressure gradients along the Mach wave characteristic directions at the rear of the shock front. The general shock refraction problem is then reduced to a single non-linear differential equation for the local shock turning angle θ as a function of upstream conditions and an unknown wave interaction parameter Φ. To lowest order in the expansion va...