Gauss-Hermite quadratures and accuracy of lattice Boltzmann models for non-equilibrium gas flows

Recently, kinetic theory-based lattice Boltzmann (LB) models have been developed to model nonequilibrium gas flows. Depending on the order of quadratures, a hierarchy of LB models can be constructed which we have previously shown to capture rarefaction effects in the standing-shearwave problems. Here, we further examine the capability of high-order LB models in modeling nonequilibrium flows considering gas and surface interactions and their effect on the bulk flow. The Maxwellian gas and surface interaction model, which has been commonly used in other kinetic methods including the direct simulation Monte Carlo method, is used in the LB simulations. In general, the LB models with high-order Gauss-Hermite quadratures can capture flow characteristics in the Knudsen layer and higher order quadratures give more accurate prediction. However, for the Gauss-Hermite quadratures, the present simulation results show that the LB models with the quadratures obtained from the even-order Hermite polynomials perform significantly better than those from the odd-order polynomials. This may be attributed to the zero-velocity component in the odd-order discrete set, which does not participate in wall and gas collisions, and thus underestimates the wall effect

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level

Experimental and analytical investigation of waterjet cleaning process

This doctoral dissertation is concerned with the development of water based cleaning technology required by industry which may substitute the traditional approach based upon the use of various chemical cleansers. The experimental study involves the waterjet removal of various coatings (rust, oil and epoxy based paints, etc.). Cleaning was carried out under a wide range of operational and geometrical conditions (standoff distance, travel speed, water pressure, diameters of sapphire nozzle and focusing tube, nozzle body type). A new designed spiral nozzle body was tested in this work. The use of surfactant was also investigated. Microscope and SEM surface were used to evaluate the degree of coating removal. The effect of various operation conditions on water consumption and cleaning rate are determined. Two new process characteristics, critical cleaning and damage standoff distances, which determine the admissible range of process variables, are first introduced in this study. The theoretical study pioneers an analytical description of waterjet cleaning. Simple equations relating the cleaning width of stationary and moving jets, which can be used to determine the optimal cleaning standoff distance, were constructed. These relations show that the maximal cleaning rate and consequently minimal water consumption can be attained at a position of 0.55-0.7 of the critical cleaning standoff distance. Experimental data substantiate the results of the theoretical study. The acquired results of the theoretical and experimental studies identify the practical range of process variables which assure complete paint removal from glass or metal surface without inducing any damage to the substrate. The spiral nozzle body was shown to provide the optimal cleaning performance. The principal result of this study, however, is a demonstration of the feasibility and effectiveness of using a high-velocity and low-volume waterjet as the single cleaning agent, and a cleanser-free technology. Also methods of development are outlined. Another major finding is the demonstration of the feasibility of using a conventional analytical description of turbulent liquid jets for the simulation of the behavior of a high speed stream of water droplets, which constitute the jets used in this study

Analytical solution of axi-symmetrical lattice Boltzmann model for cylindrical Couette flows

Analytical solution for the axi-symmetrical lattice Boltzmann model is obtained for the low-Mach number cylindrical Couette flows. In the hydrodynamic limit, the present solution is in excellent agreement with the result of the Navier-Stokes equation. Since the kinetic boundary condition is used, the present analytical solution using nine discrete velocities can describe flows with the Knudsen number up to 0.1. Meanwhile, the comparison with the simulation data obtained by the direct simulation Monte Carlo method shows that higher-order lattice Boltzmann models with more discrete velocities are needed for highly rarefied flows

Lattice ellipsoidal statistical BGK model for thermal non-equilibrium flows

A thermal lattice Boltzmann model is constructed on the basis of the ellipsoidal statistical Bhatnagar-Gross-Krook (ES-BGK) collision operator via the Hermite moment representation. The resulting lattice ES-BGK model uses a single distribution function and features an adjustable Prandtl number. Numerical simulations show that using a moderate discrete velocity set, this model can accurately recover steady and transient solutions of the ES-BGK equation in the slip-flow and early transition regimes in the small Mach number limit that is typical of microscale problems of practical interest. In the transition regime in particular, comparisons with numerical solutions of the ES-BGK model, direct Monte Carlo and low-variance deviational Monte Carlo simulations show good accuracy for values of the Knudsen number up to approximately 0:5. On the other hand, highly non-equilibrium phenomena characterized by high Mach numbers, such as viscous heating and force-driven Poiseuille flow for large values of the driving force, are more difficult to capture quantitatively in the transition regime using discretizations that have been chosen with computational efficiency in mind such as the one used here, although improved accuracy is observed as the number of discrete velocities is increased

Longitudinal Causal Inference of Cognitive Function and Depressive Symptoms in Elderly People

Objective: the association between depressive symptoms (Center for Epidemiologic Studies Depression Scale [CES-D]) and subsequent cognitive function (Mini-Mental State Examination [MMSE]) is equivocal in literature. To examine the causal relationship between them, we use longitudinal data on MMSE and CESD and causal inference to illustrate the relationship between two health outcomes. Method:  Data were obtained from the Hispanic Established Populations for Epidemiologic Studies of the Elderly. Participants included 3050 noninstitutionalized Mexican Americans aged 65 and older followed from 1993-2001. Cognitive function and depressive symptoms were assessed using the MMSE and CESD at baseline and at 2, 5, and 7 years of follow-up. Independent variables were sociodemographics, CESD, medical conditions. Marginal structural causal models were employed to evaluate the extent to which cognitive function depend not only on depressive symptoms measured at a single point in time but also on an individual’s entire depressive symptoms history.  Discussion: our results indicate that if intervention to reduce 1 points of depressive symptoms were made at two years prior to assessing cognitive function, they would result in average improvement in cognitive function of 0.12, 95% CI [0.06, 0.18],P<.0001. Conclusion: The results suggest that health intervention of depressive symptoms would be useful in prevention of cognitive impair. &nbsp