Krook Kinetic Model Research Articles

Viscous multi-species lattice Boltzmann solver for simulating shock-wave structure

The Onsager Bhatnagar Gross Krook kinetic model proposed by Mahendra et al. [32, 33] has been extended to arrive at the viscous compressible flow equations for a multi-species diffusion problem. The multi-species model ensures (i) correct exchange coefficients, (ii) positivity, (iii) that it follows indifferentiability principle and (iv) that correct amount of entropy get added as per the Onsager’s maximum entropy production principle (MEPP). A Lattice Boltzmann method (LBM) has been developed using the derived methodology within the framework of finite volume method (FVM). This solver incorporates the physics of polyatomic gases with internal degrees of freedom with flexible Prandl number. This solver can simulate the problem with species having different specific heat ratio and molecular weights. Although the equations are derived for multi-species, only binary test cases have been considered in this paper. The solver has been validated against binary shock-wave structure problems using experimental and theoretical data. Two dimensional Richtmyer Meshkov instability test case has been compared with and without viscous effects. The diffusive (viscous and mass diffusion) effects are manifested in the reduction in amplitude and increment in width of the mushroom structure. In the framework of FVM, the proposed scheme of LBM is efficient with good resolution of solutions and is easily adaptable.

A Bhatnagar–Gross–Krook kinetic model with velocity-dependent collision frequency and corrected relaxation of moments

We propose a Bhatnagar–Gross–Krook (BGK) kinetic model in which the collision frequency is a linear combination of polynomials in the velocity variable. The coefficients of the linear combination are determined so as to enforce proper relaxation rates for a selected group of moments. The relaxation rates are obtained by a direct numerical evaluation of the full Boltzmann collision operator. The model is conservative by construction. Simulations of the problem of spatially homogeneous relaxation of hard spheres gas show improvement in accuracy of controlled moments as compared to solutions obtained by the classical BGK, ellipsoidal-statistical BGK and the Shakhov models in cases of strong deviations from continuum.

Exact solution of the unsteady Krook kinetic model and nonequilibrium thermodynamic study for a rarefied gas affected by a nonlinear thermal radiation field

A development of the previous paper (J. Non-Equilib. Thermodyn. 36, 75 (2011)) is introduced. The nonstationary Krook kinetic equation model for a rarefied gas affected by nonlinear thermal radiation field is solved, instead of the stationary equation. In a frame comoving with the fluid, analytically the Bhatnager–Gross–Krook model kinetic equation is applied. The travelling wave solution method is used to get the exact solution of the nonlinear partial differential equations. These equations were produced from applying the moment method to the unsteady Boltzmann equation. Now we should solve nonlinear partial differential equations in place of nonlinear ordinary differential equations, which represent an arduous task. The unsteady solution gives the problem a great generality and more applications. The new problem is investigated to follow the behavior of the macroscopic properties of the gas, such as the temperature and concentration. They are substituted into the corresponding two-stream maxiwallian distribution functions permitting us to investigate the nonequilibrium thermodynamic properties of the system (gas particles + the heated plate). The entropy, entropy flux, entropy production, thermodynamic forces, and kinetic coefficients are obtained. We investigate the verification of the Boltzmann H-theorem, Le Chatelier principle, the second law of thermodynamic and the celebrated Onsager's reciprocity relation for the system. The ratios between the different contributions of the internal energy changes based upon the total derivatives of the extensive parameters are estimated via the Gibbs formula. The results are applied to helium gas for various radiation field intensities due to different plate temperatures. Graphics illustrating the calculated variables are drawn to predict their behavior and the results are discussed.

Investigation of the ellipsoidal-statistical Bhatnagar–Gross–Krook kinetic model applied to gas-phase transport of heat and tangential momentum between parallel walls

The ellipsoidal-statistical Bhatnagar–Gross–Krook (ES-BGK) kinetic model is investigated for steady gas-phase transport of heat, tangential momentum, and mass between parallel walls (i.e., Fourier, Couette, and Fickian flows). This investigation extends the original study of Cercignani and Tironi, who first applied the ES-BGK model to heat transport (i.e., Fourier flow) shortly after this model was proposed by Holway. The ES-BGK model is implemented in a molecular-gas-dynamics code so that results from this model can be compared directly to results from the full Boltzmann collision term, as computed by the same code with the direct simulation Monte Carlo (DSMC) algorithm of Bird. A gas of monatomic molecules is considered. These molecules collide in a pairwise fashion according to either the Maxwell or the hard-sphere interaction and reflect from the walls according to the Cercignani–Lampis–Lord model with unity accommodation coefficients. Simulations are performed at pressures from near-free-molecular to near-continuum. Unlike the BGK model, the ES-BGK model produces heat-flux and shear-stress values that both agree closely with the DSMC values at all pressures. However, for both interactions, the ES-BGK model produces molecular-velocity-distribution functions that are qualitatively similar to those determined for the Maxwell interaction from Chapman–Enskog theory for small wall temperature differences and moment-hierarchy theory for large wall temperature differences. Moreover, the ES-BGK model does not produce accurate values of the mass self-diffusion coefficient for either interaction. Nevertheless, given its reasonable accuracy for heat and tangential-momentum transport, its sound theoretical foundation (it obeys the H-theorem), and its available extension to polyatomic molecules, the ES-BGK model may be a useful method for simulating certain classes of single-species noncontinuum gas flows, as Cercignani suggested.

Exact solution of the Gross–Krook kinetic model for a multicomponent gas in steady Couette flow

The Gross–Krook (GK) kinetic model of the Boltzmann equation for a multicomponent mixture is exactly solved in a steady state with velocity and temperature gradients (Couette flow). The hydrodynamic fields, heat and momentum fluxes, and the velocity distribution functions are determined explicitly in terms of the shear rate and the thermal gradient. The description applies for conditions arbitrarily far from equilibrium and is not restricted to specific values of the mechanical parameters of the mixture. This work completes a previous study (Physica A 289 (2001) 37) based on a formal series representation of the velocity distribution function.

Non-Newtonian Poiseuille flow of a gas in a pipe

The Bhatnagar–Gross–Krook kinetic model of the Boltzmann equation is solved for the steady cylindrical Poiseuille flow fed by a constant gravity field. The solution is obtained as a perturbation expansion in powers of the field (through fourth order) and for a general class of repulsive potentials. The results, which are hardly sensitive to the interaction potential, suggest that the expansion is only asymptotic. A critical comparison with the profiles predicted by the Navier–Stokes equations shows that the latter fail over distances comparable to the mean free path. In particular, while the Navier–Stokes description predicts a monotonically decreasing temperature as one moves apart from the cylinder axis, the kinetic theory description shows that the temperature has a local minimum at the axis and reaches a maximum value at a distance of the order of the mean free path. Within that distance, the radial heat flows from the colder to the hotter points, in contrast to what is expected from the Fourier law. Furthermore, a longitudinal component of the heat flux exists in the absence of gradients along the longitudinal direction. Non-Newtonian effects, such as a non-uniform hydrostatic pressure and normal stress differences, are also present.

Shear-rate dependent transport coefficients in a binary mixture of Maxwell molecules

Mass and heat transport in a dilute binary mixture of Maxwell molecules under steady shear flow are studied in the limit of small concentration gradients. The analysis is made from the Gross–Krook kinetic model of the Boltzmann equation. This model is solved by means of a perturbation solution around the steady shear flow solution [Phys. Fluids 8, 2756 (1996)], which applies for arbitrary values of the shear rate. In the first order of the expansion the results show that the mass and heat fluxes are proportional to the concentration gradient but, due to the anisotropy of the problem, mutual diffusion and Dufour tensors can be identified, respectively. Both tensors are explicitly determined in terms of the shear rate and the parameters of the mixture (particle masses, concentrations, and force constants). A comparison with the results derived from the exact Boltzmann equation at the level of the diffusion tensor shows a good agreement for a wide range of values of the shear rate.

Influence of gravity on nonlinear transport in the planar Couette flow

The effect of gravity on a dilute gas subjected to the steady planar Couette flow with arbitrarily large velocity and temperature gradients is analyzed. The results are obtained from the Bhatnagar–Gross–Krook kinetic model by means of a perturbation expansion in powers of the external field. The reference state corresponds to the pure (nonlinear) Couette flow solution, which retains all the hydrodynamic orders in the shear rate and the thermal gradient. To first order in the gravity field, we explicitly obtain the hydrodynamic profiles and the five relevant nonlinear transport coefficients; the shear viscosity η, the two viscometric functions Ψ1,2, and the two nonzero elements, κxy and κyy, of the thermal conductivity tensor. The results show that, in general, the influence of gravity on the rheological properties η and Ψ1,2 tend to decrease as the shear rate increases, while this influence is especially important in the case of the thermal conductivity coefficient, κyy, which measures the heat flux parallel to the temperature gradient.

On the validity of a variational principle for multicomponent systems

The validity of a variational principle for nonequilibrium steady states proposed by Evans and Baranyai [Phys. Rev. Lett. 67, 2597 (1991)] is investigated in the case of a dilute binary mixture described by the well-known Groos–Krook kinetic model. We construct a perturbation solution around the unconstrained shear flow state and evaluate the phase-space compression factor, the temperature ratios, and the nonlinear shear viscosity up to the first-order approximation. All these quantities are nonlinear functions of the shear rate and the parameters of the mixture (particle masses, concentrations, and force constants). It is shown that this principle does not hold exactly, although deviations from it are small in some situations for not very large shear rates. The calculations presented here extend previous results derived for a single dilute gas.

Probabilistic approach for rarefied gas dynamics: Linearized couette flow

In the present paper, extending a new probabilistic approach for radiative transfer [1] to the linear version of the Boltzmann equation based on the Krook kinetic model, an integral equation for the one-dimensional reflection function R( τ; u, u 0) is derived from the Chapman-Kolmogoroff equation in somewhat modified form. It is assumed that the joint probability of the molecular diffuse reflection at the moving wall has the Markovian character of the multiple scattering of particles. Then, the requisite perturbation velocity distribution at the moving wall is expressed in terms of the integrated R-function. Furthermore, it is shown that, by verifying the reciprocity principle for R(τ; u, u 0) e ̄ u 0 2 , the integral equation for the reflection function S( τ; u, u 0) is derived from the integral equation for R-function in the case of the linearized Couette flow.

Heat flux induced by an external force in a strongly shearing dilute gas

Coupling between energy and momentum transport in a dilute gas subject to uniform shear flow is analyzed. Heat flux is created in the system by the action of a nonconservative external force. The results are obtained by using the Bhatnagar–Gross–Krook kinetic model. The moments of the velocity distribution function are expressed in terms of a perturbation expansion in powers of the heat field strength, the coefficients being highly nonlinear functions of the shear rate. In particular, the thermal conductivity tensor and the shear viscosity coefficient up to second-order approximation are explicitly evaluated. It is shown that the usual choice of the heat field proposed in computer simulations leads to a thermal conductivity tensor different from the one obtained in the thermal gradient problem, confirming previous results. In order to avoid this discrepancy, an alternative external force is proposed.

Shock wave structures in weakly ionized gases using the Krook kinetic model

The structures of the normal shock waves in weakly ionized nonequilibrium flows have been analyzed by the Krook kinetic models using the Maxwellian or the ellipsoidal distribution functions, and the calculated results were compared with those obtained from the bimodal approximations.

Comparison of Kinetic Theory Analyses of Linearized Couette Flow

Linearized Couette flow is considered for Knudsen numbers ranging from zero to unity using Krook's kinetic model. The velocity and stress are calculated from the numerical solution of an integral equation for the velocity. By comparison with independent calculations the relative error of these results is estimated as less than 0.2%. For the particular case of slip flow, the constant in the boundary condition is evaluated from a numerical quadrature. The accuracy of several analytic methods is compared with particular emphasis on the method of Gross, Jackson, and Ziering and that of Lees. Using previous results obtained for Knudsen numbers greater than unity it is confirmed that some form of iteration, based on an integral equation formulation, is needed to give accurate results in the nearly free molecular regime. Further, such an iteration significantly increases the accuracy of the results over the entire transition regime down to Knudsen numbers as low as one third.