---

The main purpose of this paper is to provide a mathematical model for oil future contracts. The study basis has put on the Schwartz Model (1997), but it is clear that with the small change, it is applicable for Iranian Oil Industry Futures. The Schwartz Model has considered only short jumps in the prices, but with recent oil price volatilities having significant effects on oil properties, it is necessary to include both short and long jumps in the model. According to these conditions, the model designed results in a complicated mathematical problem, which forms a Partial Differential Equation (DPE) having integral terms, and boundary and initial conditions. There is no closed solution for such equation. Thus, it should be solved by numerical methods. In the other hand, there is no access to real data for Iranian Oil Market, so we used a simulation model, but it will be possible to use an applied model in Iranian Oil Industry with real data in the future. 

---

Three-dimensional submerged jet at a sudden expansion includes chaotic hydrodynamics. At a sudden expansion, secondary flows developed adjacent to the potential core of the jet generate turbulence, and the formed eddies cause energy transfer and dissipation and decline of fluid momentum in the zone of established flow. By utilizing an efficient mathematical model of turbulence, hydrodynamic flow parameters can be predicted with a good accuracy in various locations. This paper studies the three-equation mathematical models of turbulence, namely the Walters and Cokljat (k-kl-ω), and the seven-equation Reynolds Stress mathematical model of turbulence. Comparison between the results of computational fluid dynamics using Ansys Fluent software and experimental results shows that Reynolds Stress model of turbulence predicts the results with a higher accuracy. It can be concluded that this higher accuracy is due to the use of individual transport equations for each component of the stress tensor in the normal conditions of inhomogeneous and anisotropic turbulence. Kinetic energy, very high fluid momentum and pressure fluctuations are among characteristic of a submerged jets at a sudden expansion. How the energy is dissipated by the flow and how the secondary flow structures are generated need an extensive research. In the submerged jets, because secondary flows are developed in the vicinity of jet potential nuclear and eddies are generated in various sizes, the energy is received from the mean flow and will be being dissipated while being transferred. The dissipation process can be observed during the interaction between stress and strain fields of fluid elements (second-order tensor interaction). Formation of eddies with different sizes and decay of them into smaller structures prompt the process of turbulence diffusion. The energy-bearing eddies formed in the vicinity of the jet potential core are displaced by convection terms. After these eddies are displaced, they experience decay and reduction in size (Kolmogorov microscale) and finally disappear. Rotational dynamics around the jet potential core is of a great importance in terms of flow kinetic energy dissipation; it is why the sudden expansion ratio is a number that represents the range of rotation. Therefore, understanding the flow behavior as well as how the resulting energy is generated and dissipated requires the flow parameters to be known. In order to predict the most accurate (closest to reality) values of the hydrodynamic parameters of a submerged jet, it is necessary to utilize an efficient mathematical model. Among the proposed models of turbulence, only the multi-equation Reynolds stress mathematical model has included anisotropy. Based on what have been stated so far, it seems that the existence of discrete transport equations for each component of stress tensor for a fluid and turbulence kinetic energy dissipation as well as comparison with experimental results provide the possibility of acceptable accuracy in predicting the flow hydrodynamic parameters. In this model, the term of turbulence kinetic energy generation from the mean flow, energy dissipation term, and pressure-strain term transferring the turbulence kinetic energy toward different directions of the coordinate axes are among the very important elements of the transport equation.

---

Mathematical models have the potential to provide a cost-effective, objective, and flexible approach to assessing management decisions, particularly when these decisions are strategic alternatives. In some instances, mathematical model is the only means available for evaluating and testing alternatives.  However, in order for this potential to be realized, models must be valid for the application and must provide results that are credible and reliable. The process of ensuring validity, credibility, and reliability typically consists of three elements: verification, validation, and calibration. Model verification, validation and calibration are essential tasks for the development of the models that can be used to make predictions with quantified confidence. Quantifying the confidence and predictive accuracy of model provides the decision-maker with the information necessary for making high-consequence decisions. There appears to be little uniformity in the definition of each of these three process elements. There also appears to be a lack of consensus among model developers and model users, regarding the actions required to carry out each process element and the division of responsibilities between the two groups. This paper attempts to provide mathematical model developers and users with a framework for verification, validation and calibration of these models. Furthermore, each process element is clearly defined as is the role of model developers and model users. In view of the increasingly important role that models play in the evaluation of alternatives, and in view of the significant levels of effort required to conduct these evaluations, it is important that a systematic procedure for the verification, validation and calibration of mathematical models be clearly defined and understood by both model developers and model users.