scholarly journals The structure of shock waves as a test of Brenner's modifications to the Navier–Stokes equations

2007 ◽  
Vol 580 ◽  
pp. 407-429 ◽  
Author(s):  
CHRISTOPHER J. GREENSHIELDS ◽  
JASON M. REESE
Keyword(s):  
Shock Waves ◽  
Kinematic Viscosity ◽  
Stokes Equations ◽  
Limited Range ◽  
Navier Stokes ◽  
Test Case ◽  
Viscosity Data ◽  
Hydrodynamic Models ◽  
Temperature Relation ◽  
Navier Stokes Equations

Brenner (Physica A, vol. 349, 2005a, b, pp. 11, 60) has recently proposed modifications to the Navier–Stokes equations that are based on theoretical arguments but supported only by experiments having a fairly limited range. These modifications relate to a diffusion of fluid volume that would be significant for flows with high density gradients. So the viscous structure of shock waves in gases should provide an excellent test case for this new model. In this paper we detail the shock structure problem and propose exponents for the gas viscosity–temperature relation based on empirical viscosity data that is independent of shock experiments. We then simulate monatomic gas shocks in the range Mach 1.0–12.0 using the Navier–Stokes equations, both with and without Brenner's modifications. Initial simulations showed that Brenner's modifications display unphysical behaviour when the coefficient of volume diffusion exceeds the kinematic viscosity. Our subsequent analyses attribute this behaviour to both an instability to temporal disturbances and a spurious phase velocity–frequency relationship. On equating the volume diffusivity to the kinematic viscosity, however, we find the results with Brenner's modifications are significantly better than those of the standard Navier–Stokes equations, and broadly similar to those from the family of extended hydrodynamic models that includes the Burnett equations. Brenner's modifications add only two terms to the Navier–Stokes equations, and the numerical implementation is much simpler than conventional extended hydrodynamic models, particularly in respect of boundary conditions. We recommend further investigation and testing on a number of different benchmark non-equilibrium flow cases.

An Exact Solution of the Unsteady Navier-Stokes Equations Resulting From a Stretching Surface

1994 ◽  
Vol 61 (3) ◽  
pp. 629-633 ◽  
Author(s):  
S. H. Smith
Keyword(s):  
Exact Solution ◽  
Stokes Equations ◽  
Limited Range ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Stretching Surface ◽  
Navier Stokes Equations ◽  
Short Period ◽  
Two Dimensional Flow ◽  
Surrounding Fluid

When a stretching surface is moved quickly, for a short period of time, a pulse is transmitted to the surrounding fluid. Here we describe an exact solution in terms of a similarity variable for the Navier-Stokes equations which represents the effect of this pulse for two-dimensional flow. The unusual feature is that this solution is only valid for a limited range of the Reynolds number; outside this domain unbounded velocities result.

Perturbation Procedures for Nonlinear Viscous Flows

1983 ◽  
Vol 50 (2) ◽  
pp. 265-269
Author(s):  
D. Nixon
Keyword(s):  
Perturbation Theory ◽  
Shock Waves ◽  
Transonic Flow ◽  
Stokes Equations ◽  
Viscous Flows ◽  
Two Dimensions ◽  
Experimental Results ◽  
Navier Stokes ◽  
Navier Stokes Equations

The perturbation theory for transonic flow is further developed for solutions of the Navier-Stokes equations in two dimensions or for experimental results. The strained coordinate technique is used to treat changes in location of any shock waves or large gradients.

scholarly journals Improved theory for shock waves using the OBurnett equations

2021 ◽  
Vol 929 ◽  
Author(s):  
Ravi Sudam Jadhav ◽  
Abhimanyu Gavasane ◽  
Amit Agrawal
Keyword(s):  
Shock Waves ◽  
Stokes Equations ◽  
Direct Simulation Monte Carlo ◽  
Space Velocity ◽  
Navier Stokes ◽  
Wave Flow ◽  
Complex Flow ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Mach Numbers

The main goal of the present study is to thoroughly test the recently derived OBurnett equations for the normal shock wave flow problem for a wide range of Mach number ( $3 \leq Ma \leq 9$ ). A dilute gas system composed of hard-sphere molecules is considered and the numerical results of the OBurnett equations are validated against in-house results from the direct simulation Monte Carlo method. The primary focus is to study the orbital structures in the phase space (velocity–temperature plane) and the variation of hydrodynamic fields across the shock. From the orbital structures, we observe that the heteroclinic trajectory exists for the OBurnett equations for all the Mach numbers considered, unlike the conventional Burnett equations. The thermodynamic consistency of the equations is also established by showing positive entropy generation across the shock. Further, the equations give smooth shock structures at all Mach numbers and significantly improve upon the results of the Navier–Stokes equations. With no tweaking of the equations in any way, the present work makes two important contributions by putting forward an improved theory of shock waves and establishing the validity of the OBurnett equations for solving complex flow problems.

scholarly journals Numerical simulation of compressible cavitating two-phase flows with a pressure-based solver

2017 ◽  
Author(s):  
Marco Cristofaro ◽  
Wilfried Edelbauer ◽  
Manolis Gavaises ◽  
Phoevos Koukouvinis
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Test Case ◽  
Time Step ◽  
Two Phase ◽  
Navier Stokes Equations ◽  
Liquid Phases ◽  
Flow Features ◽  
Liquid Evaporation ◽  
Semi Empirical

This work intends to study the effect of compressibility on throttle flow simulations with a pressure–based solver.The simple micro throttle geometry allows easier access for obtaining experimental data compared to a real injector, but still maintaining the main flow features. For this reasons it represents a meaningful and well reported benchmark for validation of numerical methods developed for cavitating injector flows.An implicit pressure–based compressible solver is used on the filtered Navier–Stokes equations. Thus, no stability limitation is applied on the time step. A common pressure field is computed for all phases, but different velocity fields are solved for each phase, following the multi–fluid approach. The liquid evaporation rate is evaluated with a Rayleigh–Plesset equation based cavitation model and the Coherent Structure Model is adopted as closure for the sub–grid scales in the momentum equation.The aim of this study is to show the capabilities of the pressure–based solver to deal with both vapor and liquid phases considered compressible. A comparison between experimental results and compressible simulations is presented. Time–averaged vapor distribution and velocity profiles are reported and discussed.  The distribution of pressure maxima on the surface and the results from a semi–empirical erosion model are in good agreement with the erosion locations observed in the experiments. This test case aims to represent a benchmark for furtherapplication of the methodology to industrial relevant cases.DOI: http://dx.doi.org/10.4995/ILASS2017.2017.4629

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