Solving transient free displacement flow around a sphere using DQ-IDQ

Academic thesis for receiving a master's degree in mechanics

Energy conversion trend

Abstract:

Free or natural movement heat transfer is one of the most useful phenomena in industry and in the environment around humanity. Due to its wide application, this phenomenon has attracted the attention of many researchers and has prompted researchers to follow the flow of free movement on geometries such as plates, wedges, ovals, cylinders, cones, and spheres. In the meantime, according to the events that the flow goes through until it reaches the steady state, and the importance of the flow in the transient state, this state has received the attention of researchers, while the transient flow around some geometries, including the sphere, has received less attention. Therefore, in this thesis, the transient free displacement flow around the sphere is investigated considering the magnetic field, taking into account heat absorption and production, taking into account viscosity varying with temperature and thermal conductivity varying with temperature. On the other hand, one of the most up-to-date and efficient numerical methods is the combination of piecewise differential quadratic (IDQ) with differential quadratic (DQ). Due to the newness of this method, it has been used in the numerical solution of transient conduction heat transfer problems. In this thesis, transient currents around the sphere have been investigated using this numerical method. 

1- Introduction:

One ??of the phenomena of heat transfer is free or natural movement. The change in density caused by the temperature gradient leads to fluid flow. The movement of fluid in free movement near a surface is a result of buoyancy forces, which are caused by the temperature gradient applied to the fluid near the surface and changes in fluid density. Buoyant forces that cause free displacement currents are called volumetric forces [1]. The history of the initial research that considered this flow goes back to a century ago. Since then, the data, relationships and analysis that govern this stream have grown tremendously. The innumerable interest that humanity shows in this phenomenon is a reflection of the extraordinary need that mankind felt for this interesting and vital phenomenon. The importance and diversity of the application of this phenomenon in the industry and the surrounding environment shows the wide application of this phenomenon. This phenomenon has been used sometimes alone and sometimes by combining with other transfer phenomena in heat and mass transfer.

On the other hand, considering that real physical systems or engineering problems caused by this phenomenon are described with the help of Parry equations, in most cases, solving their package [2] is extremely difficult. For this reason, numerical approximation methods are widely used to solve these equations. The most numerical methods that are used to solve such problems are finite element methods [3], finite difference [4] and finite volume [5]. These three methods are classified as low order methods. Low-order methods require a high number of computing nodes to obtain sufficient accuracy in calculations. In problems that have multiple computational dimensions, high computational capacity is needed to maintain the accuracy of calculations. Therefore, researchers began efforts to achieve methods that lead to high accuracy results with a low number of computing nodes. These methods are referred to as high order methods. Among the results of these efforts, we can mention spectral methods [6] and differential quadratic [7]. As it was said, one of the advantages of this method is to achieve the appropriate calculation accuracy while the number of calculation nodes is small. The differential quadratic method is derived from the quadratic integration method [8]. In this method, the value of the derivative of the function at each point is approximated by using the sum of the product of the function values ??in the related weight values ??along the desired direction. The key point in using this method is determining the weight coefficients. Due to the limitations in applying the initial methods of determining weight coefficients, this method was less used for many years.Until the researches conducted by researchers in the late 80s and early 90s in order to find simpler weighting coefficients, led to the introduction of this method as a powerful numerical tool in the last two decades.

With the increase in the use of this method in recent years, researchers based on the need they felt, derived other methods from the differential square method, one of these methods is single differential square[9]. This method is highly efficient in problems with highly variable gradient changes or in problems with variable boundary conditions. The idea of ??single differential quadratic method was used in 2006 in modeling waves in shallow waters. The principles of this method are based on dividing a computing domain into sub-domains and applying the differential quadratic method to each sub-domain.

In this thesis, the transient free displacement flow around the sphere is investigated by combining two differential quadratic and single differential quadratic methods.

2.1- Review of past works:

Free movement has received much attention due to its wide application in industry and in the human environment. On the other hand, considering the equations governing this phenomenon and the difficulty of presenting an analytical solution for the equations governing this flow, mankind has been forced to use numerical methods to solve this flow. On the other hand, the numerical solution of equations governing free movement has complications. The reason for this is that the momentum equation is dependent on the energy equation through the buoyancy force, and therefore the energy and momentum equations must be solved simultaneously. On the other hand, one of the factors influencing the complexity of the geometric equations is the flow on which it is checked. For example, the flow on a sphere is more complicated than the flow on geometries such as horizontal, vertical or inclined plates and even cylinders with the same condition. Garner and Gehr [1] investigated the effect of mass transfer on a non-porous sphere. Amato and Chi [2] investigated the effect of free movement around a sphere immersed in water. Broomham and Mayhew [3] investigated the flow of free movement of air over sphere.  Giola and Cornish [4] investigated the flow and heat transfer around the sphere using finite difference numerical method [10]. Singah and Hassan [5] investigated the free displacement flow around the sphere with low gravity.  Hiwang and Chen [6] investigated the effect of suction and tail on the sphere using finite difference numerical method. Chen and Chen [7] studied the free displacement flow of non-Newtonian fluid around a sphere and a cylinder using the four-order Rang-Kutta method [11]. Jafarpour and Yovanovitch [8] presented a semi-analytical solution for free displacement flow on an isothermal sphere using series. Jia and Goges [9] investigated the isothermal circumspherical free displacement flow. Nazar et al. [10] studied the free displacement flow of micropolar fluid [12] around the sphere with constant flux. They solved this problem by using color box numerical method [13]. In continuation of the previous work, Nazar et al [11] investigated the free displacement flow of micropolar fluid around an isothermal sphere using the same method as before. Mola et al. [12] investigated the effect of heat generation on the free displacement current in the magnetic field around the sphere. Cheng [13] investigated the heat transfer and mass transfer of the free displacement flow around the sphere in the vicinity of the micropolar fluid using the cubic spline collection method [14]. Beg et al. [14] investigated the effect of heat absorption and production on the free movement around the sphere inside the magnetic field which is in the porous medium. ABSTRACT: The natural convection process has developed considerable importance because of its relevance to heat transfer in many engineering applications. Due to its great applications, researchers' interest was sprang and motivated them to investigate free convection on the various geometries such as plate, cone, cylinder, sphere and etc.

Also, study of transient problems over theses geometries have its importance and attraction for investigators.