Burnett equations for simulation of transitional flows

2002 ◽  
Vol 55 (3) ◽  
pp. 219-240 ◽  
Author(s):  
Ramesh K Agarwal ◽  
Keon-Young Yun
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Small Deviation ◽  
Review Article ◽  
Navier Stokes ◽  
Space Vehicles ◽  
Surface Boundary ◽  
Transition Regime ◽  
Burnett Equations ◽  
The Continuum

Hypersonic flows about space vehicles in low earth orbits and flows in microchannels of microelectromechanical devices produce local Knudsen numbers which lie in the continuum-transition regime. The Navier-Stokes equations cannot model these flows adequately since they are based on the assumption of small deviation from local thermodynamic equilibrium. A number of extended hydrodynamics (E-H) or generalized hydrodynamics (G-H) models as well as the Direct Simulation Monte Carlo (DSMC) approach have been proposed to model the flows in the continuum-transition regime over the past 50 years. One of these models is the Burnett equations which are obtained from the Chapman-Enskog expansion of the Boltzmann equation (with Knudsen number (Kn) as a small parameter) to OKn2. With the currently available computing power, it has been possible in recent years to numerically solve the Burnett equations. However, attempts at solving the Burnett equations have uncovered many physical and numerical difficulties with this model. Several improvements to the conventional Burnett equations have been proposed in recent years to address both the physical and numerical issues; two of the most well known are the Augmented Burnett Equations and the BGK-Burnett Equations. This review article traces the history of the Burnett model and describes some of the recent developments. The relationship between the Burnett equations and Grad’s 13 moment equations as shown by Struchtrup by employing the Maxwell-Truesdell-Green iteration is also presented. Also, the recent work of Jin and Slemrod on regularization of the Burnett equations via viscoelastic relaxation that ensures positive entropy production and eliminates the instability paradox is discussed. Numerical solutions in 1D, 2D, and 3D are provided to assess the accuracy and applicability of Burnett equations for modeling flows in the continuum-transition regime. The important issue of surface boundary conditions is addressed. Computations are compared with the available experimental data, Navier-Stokes calculations, Burnett solutions of other investigators, and DSMC solutions wherever possible. This review article cites 56 references.

Beyond Navier–Stokes: Burnett equations for flows in the continuum–transition regime

2001 ◽  
Vol 13 (10) ◽  
pp. 3061-3085 ◽  
Author(s):  
Ramesh K. Agarwal ◽  
Keon-Young Yun ◽  
Ramesh Balakrishnan
Keyword(s):  
Navier Stokes ◽  
Transition Regime ◽  
Burnett Equations ◽  
The Continuum

The Burnett equations in cylindrical coordinates and their solution for flow in a microtube

2014 ◽  
Vol 751 ◽  
pp. 121-141 ◽  
Author(s):  
Narendra Singh ◽  
Amit Agrawal
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Hard Spheres ◽  
Three Dimensional ◽  
Closed Form Solution ◽  
Higher Order ◽  
Navier Stokes ◽  
Burnett Equations ◽  
Cylindrical Coordinates ◽  
Navier Stokes Equations

AbstractThe Burnett equations constitute a set of higher-order continuum equations. These equations are obtained from the Chapman–Enskog series solution of the Boltzmann equation while retaining second-order-accurate terms in the Knudsen number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Kn}$. The set of higher-order continuum models is expected to be applicable to flows in the slip and transition regimes where the Navier–Stokes equations perform poorly. However, obtaining analytical or numerical solutions of these equations has been noted to be particularly difficult. In the first part of this work, we present the full set of Burnett equations in cylindrical coordinates in three-dimensional form. The equations are reported in a generalized way for gas molecules that are assumed to be Maxwellian molecules or hard spheres. In the second part, a closed-form solution of these equations for isothermal Poiseuille flow in a microtube is derived. The solution of the equations is shown to satisfy the full Burnett equations up to $\mathit{Kn} \leq 1.3$ within an error norm of ${\pm }1.0\, \%$. The mass flow rate obtained analytically is shown to compare well with available experimental and numerical results. Comparison of the stress terms in the Burnett and Navier–Stokes equations is presented. The significance of the Burnett normal stress and its role in diffusion of momentum is brought out by the analysis. An order-of-magnitude analysis of various terms in the equations is presented, based on which a reduced model of the Burnett equations is provided for flow in a microtube. The Burnett equations in full three-dimensional form in cylindrical coordinates and their solution are not previously available.

Flow Characteristics of Microfilter in Slip/Transition Regime

2011 ◽  
Vol 694 ◽  
pp. 831-837
Author(s):  
Fu Bing Bao ◽  
Jin Xing Wang ◽  
Jian Zhong Lin
Keyword(s):  
Pressure Drop ◽  
Knudsen Number ◽  
Stokes Equations ◽  
Flow Characteristics ◽  
Navier Stokes ◽  
Transition Regime ◽  
Burnett Equations ◽  
Slip Boundary ◽  
The Difference ◽  
Slip Transition

The Burnett equations with slip boundary conditions are used in present paper to investigate the flow characteristics of microfilter in slip/transition regime. Convergent results of the Burnett equations agree very well with the results of DSMC method. The difference between the results of the Burnett and the Navier-Stokes equations increases with the increase of Knudsen number. The variation of non-dimensional pressure drop with Reynolds number and the opening factor are presented. The non-dimensional pressure drop is independent of Knudsen number.

A numerical study of the interaction between unsteay free-stream disturbances and localized variations in surface geometry

1989 ◽  
Vol 209 ◽  
pp. 285-308 ◽  
Author(s):  
R. J. Bodonyi ◽  
W. J. C. Welch ◽  
P. W. Duck ◽  
M. Tadjfar
Keyword(s):  
Numerical Solutions ◽  
Free Stream ◽  
Stokes Equations ◽  
Numerical Study ◽  
Navier Stokes ◽  
Surface Geometry ◽  
Navier Stokes Equations ◽  
S Waves ◽  
Tollmien Schlichting

A numerical study of the generation of Tollmien-Schlichting (T–S) waves due to the interaction between a small free-stream disturbance and a small localized variation of the surface geometry has been carried out using both finite–difference and spectral methods. The nonlinear steady flow is of the viscous–inviscid interactive type while the unsteady disturbed flow is assumed to be governed by the Navier–Stokes equations linearized about this flow. Numerical solutions illustrate the growth or decay of the T–S waves generated by the interaction between the free-stream disturbance and the surface distortion, depending on the value of the scaled Strouhal number. An important result of this receptivity problem is the numerical determination of the amplitude of the T–S waves.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

Numerical Solutions of the Viscous Flow Equations for a Class of Closed Flows

1965 ◽  
Vol 69 (658) ◽  
pp. 714-718 ◽  
Author(s):  
Ronald D. Mills
Keyword(s):  
Moving Boundary ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Navier Stokes ◽  
Steady Flows ◽  
Flow Equations ◽  
Steady Streaming ◽  
Difference Formula ◽  
Surface Passing

The Navier-Stokes equations are solved iteratively on a small digital computer for the class of flows generated within a rectangular “cavity” by a surface passing over its open end. Solutions are presented for depth/breadth ratios ƛ=0.5 (shallow), 10 (square), 20 (deep) and Reynolds number 100. Flow photographs ore obtained which largely confirm the predicted flows. The theoretical velocity profiles and pressure distributions through the centre of the vortex in the square cavity are calculated.In an appendix an improved finite difference formula is given for the vorticity generated at a moving boundary.Since Thorn began his pioneering work some thirty-five years ago the number of numerical solutions which have been obtained for the equations of incompressible viscous fluid motion remains small (see bibliographies of Thom and Apelt, Fromm). The known solutions are principally for steady streaming flows, although two methods have now been used with success for non-steady flows (Payne jets and Fromm flow past obstacles). By contrast this paper is concerned with the class of closed flows generated in a rectangular region of varying depth/breadth ratio by a surface passing over an open end. This problem has been considered for a number of reasons.

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