T. H. Tsang

Optimal control via collocation and non-linear programming

The collocation method meshed with non-linear programming techniques provides an efficient strategy for the numerical solution of optimal control problems. Good accuracy can be obtained for the state and the control trajectories as well as for the value of the objective function. In addition, the control strategy can be quite flexible in form. However, it is necessary to select the appropriate number of collocation points and number of parameters in the approximating functions with care.

Simulation of Condensation Aerosol Growth by Condensation and Evaporation

A new numerical method is reported for the solution of the condensation—evaporation equation, a first-order hyperbolic equation. The solution and properties of the nonlinear integrodifferential equation arising when the mass of the condensing vapor is conserved are discussed. For aerosol evolution in the conserved case it is shown that there develops an asymptotic regime analogous to the asymptotic behavior found for the coagulation process.

Comparison of Different Numerical Schemes for Condensational Growth of Aerosols

Three commonly used numerical schemes for condensational growth of aerosols are compared with analytical solutions. The sectional method is interpreted in light of the upwind differencing and the Smolarkiewicz method. Case studies for fast and slow condensation processes are used to demonstrate the performance of various numerical schemes.

On Ostwald Ripening

Results on the Ostwald ripening process of aerosols by condensation/evaporation for continuum and kinetic (free molecule) growth laws are presented. It is shown that asymptotic limit distributions are approached which agree substantially with distributions derived from similarity theory. However, the limitations inherent in previous studies of this problem are noted. The dynamics of the approach to asymptotic limit distributions are also described.

Asymptotic Behavior of Aerosol Growth in the Free Molecule Regime

Although the asymptotic behavior of aerosol growth by condensation/evaporation and by coalescence has been well established independently, no complete theory is available for such simultaneous processes. In this paper, through the use of a supercomputer, it is shown that asymptotic limit-size distributions exist for mass conserved systems when condensation/evaporation and coalescence occur simultaneously in the free molecule regime. Unlike the independent growth process either by condensation/evaporation or by coalescence, “crossover” or change from one dominating growth process to another growth process takes place continuously. It is also demonstrated that substantial savings in central processing unit time on a supercomputer can be achieved by restructuring the numerical algorithm and by vectorizing the inner DO loops.

AN EFFICIENT LEAST-SQUARES FINITE ELEMENT METHOD FOR INCOMPRESSIBLE FLOWS AND TRANSPORT PROCESSES

SUMMARY A numerical procedure based on a least-squares finite element method (LSFEM) and Jacobi conjugate gradient method (JCG) is presented for the numerical solution of fluid flow and transport problems. Unlike many finite element methods, the LSFEM does not involve any upwinding factor. Furthermore, the LSFEM leads to a symmetric and positive definite linear system of equations which can be solved satisfactorily by a preconditioned conjugate gradient method. Four examples, lid-driven cavity flow, thermally-driven cavity flow, Rayleigh-Benard convection and doubly-diffusive flow, are presented to validate the preconditioned conjugate gradient method. A comparison of the least-squares finite element method and theGalerkin finite clement method (GFEM) is also given. Finally, we demonstrate that the least-squares finite element method with the Jacobi conjugate gradient iterative technique is a promising approach to solve three-dimensional fluid flow and transport problems.

On a Petrov-Galerkin Finite Element Method for Evaporation of Polydisperse Aerosols

A Petrov-Galerkin finite element method is derived for evaporation of polydisperse aerosols. It is demonstrated that this numerical method is accurate and computationally efficient. Together with an appropriate grid system and upwinding factor, this method can reduce spurious oscillations to a negligible level and provide reliable results for a wide range of initial size distributions and evaporation rates. Its performance is superior to the upwind differencing method and the sectional method. It is also shown that analytical solutions are very useful in a priori design of grid systems for simulations of realistic aerosol systems, and evaporative cooling is a significant factor in modeling evaporation of polydisperse volatile aerosols.

Effect of Evaporation on the Extinction Coefficient of an Aerosol Cloud

A novel numerical scheme is devised for the solution of evaporation of aerosol clouds. This scheme combines the salient features of the Galerkin Finite Element Method and the positive definite method of Smolarkiewicz. It can allow the use of large time steps, and the numerical solutions remain stable even after prolonged simulations. The numerical results compare favorably with the analytical solutions for different continuum evaporation rates. The effects of the volatility of the compound and the initial particle size distribution on the evaporation process are demonstrated. It is also shown that evaporation leads to a significant decrease in the total extinction coefficient of the aerosol cloud.

On First-order Formulations of the Least-squares Finite Element Method for Incompressible Flows

To reduce the element continuity requirement for the least-squares finite element method (LSFEM), it is customary to rewrite the Navier–Stokes equations as first-order partial differential equations. In this work, numerical experiments for three-dimensional incompressible flows are carried out by using the LSFEM based on three types of first-order formulations, namely the velocity–vorticity–pressure (VVP) formulation, the velocity–stress–pressure (VSP) formulation and the velocity–velocity gradient–pressure (VVGP) formulation. In addition, two modifications for each formulation are considered. Numerical results indicate that proper problem formulation can significantly reduce computing time and improve the accuracy of numerical solutions.

Effect of temperature, atmospheric condition, and particle size on extinction in a plume of volatile aerosol dispersed in the atmospheric surface layer.

The objective of this work is to study the effects of ambient temperature, atmospheric condition, and particle size on the extinction coefficient of diesel fuel and fog oil smoke. A first-order closure model is used to describe the turbulent diffusion of the smoke in the atmospheric surface layer. Mean values of wind speed and diffusivity in the vertical direction are obtained by the use of the Monin-Obukhov similarity theory. The 2-D crosswind line source model also includes the aerosol kinetic processes of evaporation, sedimentation, and deposition. Numerical results are obtained from simulations on a supercomputer.