Abstract: Nowadays, data center industry faces strong challenges for increasing the speed and data processing capacities while at the same time is trying to keep their devices a suitable working temperature without penalizing that capacity. Consequently, the cooling systems of this kind of facilities use a large amount of energy to dissipate the heat generated inside the servers, and developing new cooling techniques or perfecting those already existing would be a great advance in this type of industry. The installation of a temperature sensor matrix distributed in the structure of each server would provide the necessary information for collecting the required data for obtaining a temperature profile instantly inside them. However, the number of temperature probes required to obtain the temperature profiles with sufficient accuracy is very high and expensive. Therefore, other less intrusive techniques are employed where each point that characterizes the server temperature profile is obtained by solving differential equations through simulation methods, simplifying data collection techniques but increasing the time to obtain results. In order to reduce these calculation times, complicated and slow computational fluid dynamics simulations are replaced by simpler and faster finite element method simulations which solve the Burgers‘ equations by backward, forward and central discretization techniques after simplifying the energy and enthalpy conservation differential equations. The discretization methods employed for solving the first and second order derivatives of the obtained Burgers‘ equation after these simplifications are the key for obtaining results with greater or lesser accuracy regardless of the characteristic truncation error.
Abstract: Controlling the flow of fluids is a challenging problem
that arises in many fields. Burgers’ equation is a fundamental
equation for several flow phenomena such as traffic, shock waves,
and turbulence. The optimal feedback control method, so-called
model predictive control, has been proposed for Burgers’ equation.
However, the model predictive control method is inapplicable to
systems whose all state variables are not exactly known. In practical
point of view, it is unusual that all the state variables of systems are
exactly known, because the state variables of systems are measured
through output sensors and limited parts of them can be only
available. In fact, it is usual that flow velocities of fluid systems
cannot be measured for all spatial domains. Hence, any practical
feedback controller for fluid systems must incorporate some type of
state estimator. To apply the model predictive control to the fluid
systems described by Burgers’ equation, it is needed to establish
a state estimation method for Burgers’ equation with limited
measurable state variables. To this purpose, we apply unscented
Kalman filter for estimating the state variables of fluid systems
described by Burgers’ equation. The objective of this study is to
establish a state estimation method based on unscented Kalman filter
for Burgers’ equation. The effectiveness of the proposed method is
verified by numerical simulations.
Abstract: In this paper, a backward semi-Lagrangian scheme
combined with the second-order backward difference formula
is designed to calculate the numerical solutions of nonlinear
advection-diffusion equations. The primary aims of this paper are
to remove any iteration process and to get an efficient algorithm
with the convergence order of accuracy 2 in time. In order to achieve
these objects, we use the second-order central finite difference and the
B-spline approximations of degree 2 and 3 in order to approximate
the diffusion term and the spatial discretization, respectively. For the
temporal discretization, the second order backward difference formula
is applied. To calculate the numerical solution of the starting point
of the characteristic curves, we use the error correction methodology
developed by the authors recently. The proposed algorithm turns out
to be completely iteration free, which resolves the main weakness
of the conventional backward semi-Lagrangian method. Also, the
adaptability of the proposed method is indicated by numerical
simulations for Burgers’ equations. Throughout these numerical
simulations, it is shown that the numerical results is in good
agreement with the analytic solution and the present scheme offer
better accuracy in comparison with other existing numerical schemes.
Abstract: A fully implicit finite-difference method has been proposed for the numerical solutions of one dimensional coupled nonlinear Burgers’ equations on the uniform mesh points. The method forms a system of nonlinear difference equations which is to be solved at each iteration. Newton’s iterative method has been implemented to solve this nonlinear assembled system of equations. The linear system has been solved by Gauss elimination method with partial pivoting algorithm at each iteration of Newton’s method. Three test examples have been carried out to illustrate the accuracy of the method. Computed solutions obtained by proposed scheme have been compared with analytical solutions and those already available in the literature by finding L2 and L∞ errors.