Zero dissipation limit to a Riemann solution consisting of two shock waves for the 1D compressible isentropic Navier-Stokes equations

2013 ◽  
Vol 56 (11) ◽  
pp. 2205-2232 ◽  
Author(s):  
YingHui Zhang ◽  
RongHua Pan ◽  
Zhong Tan
Keyword(s):  
Shock Waves ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Riemann Solution

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 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.

scholarly journals PREDICTION OF TRANSONIC BUFFET ONSET FOR FLOW OVER A SUPERCRITICAL AIRFOIL- A NUMERICAL INVESTIGATION

2013 ◽  
Vol 43 (1) ◽  
pp. 48-53 ◽  
Author(s):  
Arif Abdullah Rokoni ◽  
A.B.M. Toufique Hasan
Keyword(s):  
Shock Waves ◽  
Stokes Equations ◽  
Lift Coefficient ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Supercritical Airfoil ◽  
Aerodynamic Properties ◽  
Sst Turbulence Model ◽  
Shock Oscillation ◽  
Transonic Buffet

Transonic flow over a supercritical airfoil leads to the appearances of unsteady shock waves in theflow field. At certain flow conditions, the interaction of unsteady shock waves with boundary layer becomescomplex and generates self-excited shock oscillation, lift fluctuation and thus initiate the buffet. In the presentstudy, Reynolds averaged Navier-Stokes equations with k-? SST turbulence model has been applied to predictthe shock induced buffet onset for the flow over a supercritical airfoil NASA SC(2) 0714. The free streamtransonic Mach number is kept in the range of 0.71 to 0.75 while the angle of attack is varied in a wide range.The onset of buffet is confirmed by the fluctuating aerodynamic properties such as lift-coefficient, pressurecoefficient, static pressure and so on. The self-excited shock oscillation and the corresponding buffet frequencyare numerically analyzed.DOI: http://dx.doi.org/10.3329/jme.v43i1.15782

On the Numerical Solution of Two-Dimensional, Laminar Compressible Flows With Imbedded Shock Waves

1972 ◽  
Vol 94 (4) ◽  
pp. 765-769 ◽  
Author(s):  
W. D. Goodrich ◽  
J. P. Lamb ◽  
J. J. Bertin
Keyword(s):  
Shock Waves ◽  
Compressible Flows ◽  
Stokes Equations ◽  
Numerical Technique ◽  
Leading Edge ◽  
Navier Stokes ◽  
Surface Boundary ◽  
Navier Stokes Equations ◽  
Numerical Experimentation ◽  
Explicit Finite Difference

The complete, time-dependent Navier-Stokes equations are expressed in conservation form and solved by employing an explicit finite difference numerical technique which incorporates artificial viscosity terms of the form first suggested by Rusanov for numerical stability in the vicinity of shock waves. Surface boundary conditions are developed in a consistent and unique manner through the use of a physically oriented extrapolation procedure. From numerical experimentation an extended range for the explicit stability parameter is established. Also employed is an additional convergence parameter which relates incremental spatial steps. Convergence of the transient solution to a steady state flow was obtained after 400 to 500 time steps. Sample solutions are presented for supersonic flow of air over the leading edge of a slightly blunted flat plate, past a backward facing step, and in the near wake of a blunt trailing edge. Free-stream Mach numbers from 2 to 10 are included in the sample computations.

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